Can getting an access token be simplified for the clients using my API app solution?

While there isn't a way to simplify the authentication, captcha, and authorization process (though the captcha isn't always shown), this process only needs to be done once per user. After that, the app can store and re-use the resulting credentials without reprocessing that flow.

In order to maintain long-term access to an account via the API without having to manually re-authorize the app repeatedly, the app should have the user process the OAuth app authorization process once, but request "offline" access when doing so.

When requesting offline access, the app will receive a "refresh token". The app can store and re-use that refresh token repeatedly in order to retrieve new short-lived access tokens, whenever needed. This process of using a refresh token to get a new short-lived access token can be done entirely programmatically, and does not require manual user interaction. Refresh tokens don't expire (though they can be revoked on demand).

You can find more information on this process in the OAuth Guide and authorization documentation. If you're using an SDK, be sure to refer to the respective SDK documentation as well.