Yes, ‘Gabba’, agree. We always ask if we haven’t already lined up a lease car, (unless it emphatically states a car is needed); we proffer International licence verification, explain how insurance is easy to be adapted and hope the Hosts have read in our reviews, how well we have cared for cars on previous Sits. Naturally we always offer to drive our Hosts to airports, etc.
If dogs have to be driven to parks, exercise areas or appointments, we do expect a ‘doggie’ car as often rental companies put a restriction on animals in their vehicles. This is always noted early in our application contact.
We take things as they come too, it’s part of the fab. experience of Housesitting!
As an example, recently a Host asked if we could drive them to the airport (an hour plus away) then collect on return with late, and very early, times. We were a tad disappointed that her car sat in the garage, unoffered, so we had to use our car with no Host car key available either (we just assumed there would be!) for set Vet/Groomer consultations during our stay.
A taxi quoted 125€-150€ EACH trip… on return, we were presented with a box of Shortbread! Just the offer of a little petrol reimbursement would have been appreciated! (Ironically, we’d possibly have declined except, unfortunately, there were other posted features and amenities ‘missing’ too… 40’C+ heat for several weeks and the air conditioning ‘forgotten’ to be reconnected after renovations was unacceptable and discourteous to Sitters).
Please note… this was an unusual occurrence as the vast majority of our Hosts from our 55+ Sits, have been sensational and brilliantly generous!
PS…The Shortbread was delish!!
Hope you reviewed the host honestly.
I’m glad you enjoyed it! And I’m astonished you don’t feel it compensated your trips to the airport. I’d do anything for a box of shortbread! 
@Tasmazias You sound like very honourable people who go the extra mile as sitters. But I feel you were taken advantage of by those hosts. They should have given you their own car- for the airport runs at least- or offered petrol money for you using your own car.
We actually withdrew from a sit - booked 3 months ahead- when the hosts casually requested- 2 weeks before the sit- that we drive them to the airport and pick them up on their return at 1.30am!! This was for a 10 day sit in a country where airport parking is very cheap but its 1.5 hours each way to drive there. They said ofcourse we’d drive in their car but did not offer it for use during the sit. We had our own car so that part was no advantage to us anyway. But the unfriendly timings and their expectations for us to act as free chauffeurs made us feel used. They were shocked by our reaction and said they’d always be happy to take sitters to/from the airport!! That is a completely different scenario if the sitter is flying in especially for the sit! These wealthy hosts just wanted to save money on airport parking and squeeze an extra duty out of their sitters!
I honestly cannot and will not sit for entitled hosts. It has to feel like a fair exchange- as far as is possible.
In another scenario we were once invited down to the south of France for a lovely villa/pool sit. We were offered use of car in exchange for dropping each other to and from the local airport. This felt very fair and saved us all costs and hassle. We didn’t have to rent a car for 3 weeks and they didn’t need to take a taxi or pay airport parking. Win win- as it should be. 
Also your hosts not bothering to organise the AC properly in time for your sit is plain disrespectful.
I do hope you mentioned these negative things in your review.
We are fighting an issue now where our sitter used our truck with permission to take our dogs for a hike in a nearby state park rather than their rental car. They decided to take our truck on their vacation explorations, without our permission, and proceeded to fill the diesel tank with gas. They promised to pay until they had to pay the tow fee and the repair fee. It’s a mess right now. I do not recommend loaning a vehicle AT ALL!
That is so sad to read. As a sitter I would never make such demands and would read the owners profile carefully to see if it or I were suitable. I might ask questions but I would never make demands. I’m quite shocked!
The OP asks “Should the homeowner provide a vehicle?”
Short answer: no. There is no “should” about it.
As sitters we had one HO let us use their car to take their dog out to their fave spots.
We’ve also had various sits with golf cart use (which we love!)
Another HO said it would be ok since their cat had a medical emergency during our sit last year.
Currently, if a HO feels comfy with us using their car, great. If not, that’s cool too.
We don’t let sitters use our car though. We usually drive to and from and only have one car that comes with us on trips.
For anyone letting folks use their car:
Get pic of valid Driver’s License from sitters & Proof of insurance.
IMO a car is not needed for sits.
Some exceptions;
If the host require the sitter to drive pet to other locations, then a vehicle should be provided as it is the need of the host, not the sitter.
If rural, the host should have a plan for how to handle emergencies for transport of the pet (could be a neighbor that could drive to vet, an account set up at the vet for house call or :other:)
For insurance purposes, an option could be to cover the cost of a rental - then issues would be between sitter and rental firm, but the problem is often that transport of pets isn’t allowed for rentals. So that would need to be checked.
For rural sits it can be difficult to attract sitters, as many come by air and such, and many sitters are not willing to pay for a rental in addition to travel costs. It could then be favorable to offer a car to attract applications / sitters. But again, that is the need of the host, not the sitter.
If a host consider to offer a car it could well be another car than they themselves drive. I’ve seen offered simpler work vehicles for instance. But it should ofc be safe, approved, registered - and transparent before sit what it is.
Positive observation:
Like many threads on this forum, this is a great one that any new HO or sitter could read through to get an excellent sense of what the consensus is on both sides as far as what’s acceptable.
And that’s what newcomers should be doing. Educating themselves.
To add my experience as an HO, my listing makes it clear that there’s no car included and I’d say it’s only perhaps one out of every ten applicants I talk to that will show they’ve clearly not read the listing. Most have been just fantastic.
(In my decision-making process I’ve taken into account the desirability of my location and sitter interest vs my willingness to deal with car-related incidents. Even if insurance were to cover damage, it would be my time… hours and hours that I do not have… spent sorting out claims, repairs, etc etc. Important to also take into account your emotional well-being and what you have the mental wherewithal to deal with.)
Bottom line, as others have said above, the sit is supposed to be an even exchange; good accommodations for good pet care.
Sitting or hosting should neither be viewed as a work contract nor a favor of any sort. It’s a barter. We all save some money (in theory/when done right). And/or we all get the added benefit of peace of mind when there are detailed reviews.
Posts above like Garfield’s detail noteworthy exceptions/considerations to the whole car thing.
(Like many HOs I like to do little extra things to make my sitters feel welcome and appreciated, but these are always received as icing on the cake, as a surprise… not as payment, not expected or demanded.)
Regardless of how THS advertises and what perceptions (or misperceptions) said advertising may lead to, one thing is true in any forum or setting (or country): there’s good and bad everywhere.
Americans, Europeans, and every other population is the same… mostly lovely, but alas, there will always be some that are not.
Likewise, some human beings have common sense, are socially adept, have a certain degree of emotional intelligence… but these are not things that come easily to everyone in the same way.
Doesn’t matter whether it’s on this forum or another… or in business or social interactions…
Here, and in life in general, it’s all about learning… and spotting red flags before they become a problem (your problem).
If this community and the world in general needs anything I find it always boils down to 2 things: compassion and critical thinking.
So yes, please have compassion for those who don’t find human interaction so easy… but use your critical thinking skills to ensure your physical/mental wellbeing/safety if you’re a sitter. And the safety of your home, person, and pet if you’re the HO.
That generally means saying “no” (or not engaging) when you come across someone who doesn’t know how to be a reasonable human being. (And… I’m finally going to go ahead and say this: No, it’s not your job to teach a random grown *** adult how to be one. That’s an expectation I’ve come across in messaging/applications and on the forum, and… just no. But also, as I said earlier, the majority of my interactions here have been stellar.)
Again, reading through a thread like this gives an excellent sense of what the consensus is/should be regarding appropriate behavior on both sides, so today I found myself just really appreciating this conversation and all the thoughtful responses above.
Overall my sitters found through here have been a blessing in my life and the life of the pet I love so much that I dedicate hours (and hours and hours) to all this!
Listings, forums, and applications, oh my! 
@KChev Every time I’ve had that happen, the HO drive if they are in the car. It’s easy enough to jump into the driver’s seat after I’ve helped them with their bags. I feel more comfortable letting them drive their own car.
Dan
Based on your post I was so tempted to start a thread titled, “My nice, boring, normal sit where nothing forum-worthy happened.” And then I realized - the reviews are already THAT! There are thousands of reviews on THS of sits that went JUST FINE.
But I have to admit, it is fun sometimes reading the crazy here on the forum. 