From my understanding, being in the YPP is a requirement for using the YouTube Analytics API as the channel has to be eligible for monetization, which is what the YPP program is for. The ability to link doesn't even pop up without being in the program. I am not aware of another alternative for non-YPP channels.

Since your channel is not linked to your Google Cloud project, you will get the 403 error as the project does not have permission to access the channel's analytics data without the linkage.

As you mentioned, you are able to use the YouTube Data API because this doesn't require project linking. The Data API can provide some basic publicly available metrics, such as views, likes, and comments for videos and subscriber count and total views for the channel.

See: developers.google.com/youtube/analytics