You are very fortunate, we have experienced several.
We have also taken our friend’s border terrier on several car trips and he was fine until one day it started to rain and we put on the rear wiper and he went ballistic.
Yes, not all dogs can travel in a car. A friend’s dog suffers from severe car sickness
Yes, some get car sick, some are anxious and/or freak out. And some hosts might not want their dogs going on rides. That’s why I always discuss with hosts first. Not a big deal for me.
The key thing in the original post: Some hosts assume that sitters will automatically be open to taking their dogs on rides, without compensation for gas or such. To me, that kind of entitlement is not cool. I don’t even care about gas money, but hosts shouldn’t be entitled. Some sitters want gas money and some don’t want to take pets in their cars.
There’s a big difference with a sitter choosing to take a dog in their car to a place that the sitter wants to visit .
We do lots of sits like this where we can go and visit a different beach , park, forest, river each day at a time that is convenient for us and safe for the dog . We also can walk from the doorstep without using a vehicle on the days when we want to stay local .
That is entirely different scenario to what the OP describes where a host requires that you MUST take the dog in your vehicle each day .
For example we saw a sit that required the sitter to take not 1 but 5 dogs to a dog park at a specific time slot each and everyday of the sit . There was no flexibility in the arrangement . It was not unsurprising that they did not get any sitters apply .
when a sitter has to transport a pet in their own car and. pay for the gas etc, that could be the deciding point betweeen a free exchange and a paid one.