Hi James
That is actually quite a good idea, which I can see advantages in for both sides. I travel using Airbnb and I host on Airbnb. However, I must point out the hoster's side, it is incredibly difficult to manage a listing or listings, as far as prices are concerned. I also know a lot of hosts don't want to let out their properties for less than a fixed price as it does involve quite a lot of preparation work, especially if you have 1 or 2 day lets all the time. It would also lead to a lot of last-minute bookings, which if you do meet and greet instead of self check-in and out, could be a logistical nightmare.
Travellers always have the opportunity to ask a host if they would consider dropping the price for a last minute let if you are really interested in the property. Hosts have the option to make a special offer, so all you need to do is enquire, rather than book straight away. I did this for a 5 week booking when I went to Istanbul. It worked fine, I got my reduction for the longer stay. So in a way the sytem is there, just the onus of the work is on the traveller and not on the host. Try it, I'm sure if you offer a sensible price a host won't decline if it's a last minute booking. 😉