Look at the attack vectors that are actually being used, and address them specifically, with minimally invasive measures.
If the problem is apps that allow remote control of your device, that people can be socially engineered into installing, put up barriers to gaining just that permissions. That approach would actually help motivate the problem (as scammers can now just use Google-approved apps for such things).
If the problem is ads that are pushing scams, Google could start with eradicating them from their own network. They seem to be the primary source. And, god forbid, perhaps even offer an ad blocker integrated in Android. (Yeah, I know.)
If the problem is scammers pretending to be a friend or family member in need of help though social apps, Google could force these apps to help users identify these cases (using local privacy friendly heuristics is course) for inclusion in the Play Store. And no, they wouldn't be able to demand the same from apps installed from elsewhere, but that should be firmly outside of their sphere of responsibility. And casual users would be extremely like to stick with the default app store anyhow.
Note that all three of these proposals provide a measure of safety from the problems they are addressing much larger than what Google is attempting by banning all non-Google-authorized applications.