How would the user get the long password into the mobile device's password manager in the first place?
They would sync their encrypted password database to the mobile device. Alternatively, the password manager could generate the long password itself on the device if that is where the account is being created.
Provided the user has an own PC. Good luck logging in at a public library or Internet cafe.
The fundamental problem with this scenario is that you're proposing to place your trust in a public PC you can't control. At a minimum, that particular login session must be consider potentially compromised no matter what authentication scheme you use. Having said that, there are some options if you're forced into this scenario. An OTP hardware token would be preferred; at least that way an attacker can only hijack the current session, rather than having the means to sign in as you in the future. If you do use a traditional password then it must be considered compromised and should be changed from a secure PC as soon as possible.
And store this "own distinct, revocable API key" in what secure manner? Client applications distributed as free software have already run into problems with how to store an OAuth 1.0a or 2.0 client ID and client secret.
The problem you're referring to relates to application-level keys which are meant to identify the developer of the application rather than the user. The only real solution in such cases is to make your app communicate with one of your own servers, which holds the API keys and performs API access on behalf of the app. Any keys distributed with an app (whether open source or proprietary) must be considered compromised.
In this case the API key is user-specific, not app-specific, so there is no distribution issue. The user logs in and generates an API key, which the application then stores for future use. The API key is the application's password, permitting limited access to the user's account. (For example, it should not be possible for an app to change the account password or generate additional API keys using an API key.)