The same kind of growth that any new, deep pocketed entrant in the hospitality industry might have - it would just have to come at the expense of incumbents.
In 30 years, an investor like that may hope that AirBnB might look more like Hilton Worldwide, then what it is today.
The AirBnB network has ~3,000,000 suites across 60,000 cities. Even if they were to comply with all local laws, they would still have margins, and room to grow, and would weather downturns better then traditional hotel chains ('Sharing economy' means that they aren't stuck paying mortgages on rooms they can't rent during a recession.) They can also expand into the traditional hotel chain business, by building or buying properties.
I'm not sure I understand, why is this not the case?
The practice of franchising is popular within the hospitality industry among most major hotel chains, as the parent company does not have to pay for the maintenance and overhead costs of franchised properties.
[1]https://en.wikipedia.org/wiki/Hilton_Worldwide#Franchising
Your own source says that 70% (by rooms) was franchised as of 2013, which suggests 30% is Hilton owned.