If Airbnb started today, which platform (web or mobile) should they start with and why?

I'm not sure if I understand your question. But if anyone one started something on a budget, the best choice (temporarily) would be leveraging Wordpress.
Typically Wordpress is scalable up to about 10-15k total users... By then you should be working on your own platform and bigger programming team.

If you're not on a budget the question is almost irrelevant.

I would say they would start with a website, for a few reasons:

- You need to build a web platform for mobile apps to communicate with, you might as well build a frontend for it too.

- Responsive frameworks and technology is very advanced, you could go with a mobile-friendly approach and cover many devices & browsers from day 0.

- Easy to validate, gather feedback and tweak.

- Once they have traction and a large user base, start targeting specific mobile platforms and build apps for them. You take the guesswork out of defining what/if yo build a mobile app.

That depends on the market, they should have use the usage reports for starting countries. Some countries have mind-blowing rates of mobile usage (some countries have about 3% desktop devices)