To stop scammer-guided malware installation, and probably those "download whatsappupdate.apk for free new emoji" ads that pop up all the time.

Google doesn't care about F-Droid one way or the other. It's a niche project that barely registers on the scale of all Android users.

They don't care about F-Droid but they do care to choke out any potential competitors to their ecosystem before they can get a foothold. See their behavior surrounding device certification for example. They want to abuse the network effects of their ecosystem to prevent consumers from leaving. This is just more of that - vendor lock-in masquerading as an unfortunate necessity.