Exactly.

If my device is compatible, does it automatically have access to Google Play and branding?

No. Access isn’t automatic. Google Play is a service operated by Google. Achieving compatibility is a prerequisite for obtaining access to the Google Play software and branding. After a device is qualified as an Android-compatible device, the device manufacturer should complete the contact form included in licensing Google Mobile Services to seek access to Google Play. We’ll be in contact if we can help you.

https://source.android.com/docs/setup/about/faqs

Google services are entirely missing from Android open source. The Google Play package is what contains the entirety of Google’s services.

Not sure if anyone remembers but back when cyanogenMod was the go-to, early versions had Google services included. Google sent a cease and desist notice and said it was a license violation. You cannot distribute it as part of the OS by default. The next release of cyanogenMod had it removed. Users had to flash the package if they wanted it.