Hey @razorfrog – I was able to reproduce this issue with Site Kit as well. I set up Site Kit on around 28 sites using the same Google account over a period of a couple of days and eventually lost access to data on the earlier sites, requiring that I re-login to regain access. As you anticipated, that same API limits affect your account access via Site Kit, since these are account/API limits.
Am I correct in guessing that you are setting up or connecting a new Google analytics property for each site? Are these for sites being built for clients? If so, my recommendation would be to share access to the site property with the client directly in Google Analytics, then have them go thru the Site Kit auth flow in WordPress to access their site data.
You can use a separate WordPress login to initially log in and configure Site Kit, then have the client log in and use their Google account to access the (preconfigured) data. This way only their account connection needs to remain intact, and the fact that you are setting up multiple sites with your account wont matter.