Its hard to know exactly what the privacy implications will be of installing Play Service and Play Store on the device.
A lot depends on the apps that are in the user profile and how you have things configured - which apps have which permissions. The only apps that will share data with Play Services have Google libraries running inside themselves.
The libraries for Googles crash reporting, analytics and advertising services, as long as the app has network, will connect to Google directly if there is no Play Services on the device.
If you arent using a VPN you may be concerned about the Google apps being able to see the different IP addresses you connect from. But theres the same concern for their libraries running in other apps. You could reduce this by constraining their ability to run in the background. But this would also impact the ability to get instant notifications for any messenger apps.
Its also worth considering that theres many other companies that make libraries, designed to make app development easier, which are also included in many apps. Some of these libraries are purposely designed to also collect as much data as possible for sale to data aggregation companies. Also worth noting that libraries within an app inherit the permission that the app has been granted!
Its also possible to get apps and update them via Neostore, FFupdater, Accrescent. With Neostore its worth considering which repos you have enabled, but theres a whole load to choose from.