Why would airbnb not be happy about this? They would really only be gaining money as their income is proportional to the owner's income. (Else this product doesn't help anyone, including the owner of the property)
The same reason Craigslist didn't like Padmapper scraping all their listings: The listings bootstrap a competitor who will later try to replace them. Example:
1. Get sellers used to visiting www.beyondpricing.com instead of www.airbnb.com - you don't have any problems attracting sellers, because you can get their listing in front of as many people as airbnb.
2. Add some sort of multi-site listing search or something, to get buyers also visiting you directly instead of going via airbnb. Advertise your site on your sellers' airbnb listings.
3. Once buyers and sellers are used to using your site instead of airbnb, launch your own listings service and cut airbnb out all together. (Or better yet, launch your own listings and only invite the profitable low maintenance buyers and sellers with established histories, leaving the dregs to airbnb)
This allows you to sidestep the problem of "sellers don't want to use us as there are no buyers, buyers don't want to use us because there are no sellers".
This actually seems like something Airbnb would want to acquire. From what I've heard, setting the price is still one of the most painful parts of hosting. Having this available for all users by default would be a huge plus.
Beyond Pricing is free during beta, and then we will charge a reasonable monthly rate per property, similar to other SaaS solutions for vacation rentals. The channels like Airbnb actually love it because it helps maximize revenue across the system, making the pie from which they take a 10% commission on larger.
I'd be careful about assuming the platforms will "love it". AirBnB, in particular, is very sensitive to customer experiences and ever-changing prices are known to be extremely annoying. A delicate balance must be struck.
Maybe there could be an option for "max volatility" or something where an owner would give it a percent and say "Don't change my price by more than x% in y time".