It’s great that you are looking to improve this feature. I think it could be a really powerful tool to help us find sits.
Currently we can only have 3 saved searches which is not enough for us as full time housesitters.
As @Provence said it would be useful to have better location filters - whether that is continents or even just be able to have multiple locations. I may have already mentioned this before, but RightMove have a great feature on their website where you can draw on the map the area you want to search and it returns results only within that area.
Another feature would be to just set a start date. Sometimes we have a few months booked up but nothing after. So a search filter starting from the start of our gap running continuously would be nice. Obviously we could just scroll through the calendar and enter an end date far in the future, but would be handy to just have open ended.
On the form you mentioned adding to dashboard, I think this would be good, currently it takes up quite a large space on the search page and if you are extending the ability to have more than 3 it might be too big for there. It might be nice to have alerts saying when new matches have been found on the website as well as the app.