“…an inferior browsing experience” FTFY

The port of entry that originally denied you entrance immediately notes that in the system and you are “Flagged” at all other ports of entry.

As I understand it, LineageOS does have Google Play Services built in. The microG version is for if you wish to spoof those services for apps that rely on them if you choose to not use actual google background services (location, play store etc)

You can use a Privacy card with any name you want on it although some small vendors with actual people running the purchase process can read and take issue when the “card” is supposedly issued to a “Mybig Blackdog”