The problem with webapps using a BAAS intended for mobile apps is the same origin policy. I've yet to find any providers that support Cross-Origin Resource Sharing. This restriction doesn't apply to mobile apps (or Chrome extensions)
The primary client for an API I'm currently working on is an iOS app, so for the web version of the client, it just made sense to build on top of the same OAuth-authenticated API. Instead of traditional sessions, we store the access token in a cookie and sign each authenticated request just as you would any third-party API.
The BaaS architecture is actually a perfect situation for making privileged calls with CORS, because the server is wholly responsible for the user's identity and permissions.