Uber does offer an API, but I think part of the problem is that a lot of these providers either worry about becoming interchangeable, or believe - rightly or wrongly - that their brand is strong enough not to want to let intermediaries cut in.
So I think you're right in as much as that at least in some areas it'll take providers big enough to say "if you don't work with us all of our millions of customers will see your competitors first".
A smaller player will likely have alto start small and be content to not have as broad coverage as they'll like.
Maybe couple with papering over some with human intervention like more traditional concierge services until they can go to providers with proven demand.