It hasn't always been allowed within our rules, and indeed isn't allowed by the rules of most other functioning democracies.

To the extent that the US system of government is more influenced by money than it is by votes, it is plutocratic rather than democratic.

Except every other candidate accepts donations and has to be more transparent. Every other serious candidate is appearing on the debate stage and town halls.

Bloomberg is buying off people and hasn't been forced to defend his atrocious record, at all.

and plenty of people support aggressively campaign financing reform and public funding of elections.

Well, there’s a lot of regulation around campaign finance: spending limits, disclosure requirements, etc. As far as I know, Bloomberg doesn’t haven’t to adhere to any of that, because it’s not campaign finances, it’s his own personal (massive) wealth.

I think this practice is a bit worse than other forms of electioneering since it is not clear if views being expressed come directly from the candidate or not (unlike a TV ad or a rally). When governments do something similar this, their methods are rightly criticized as being undemocratic [1,2]. I think a similar line of reasoning applies to US election politics.

[1] https://en.wikipedia.org/wiki/50_Cent_Party

[2] https://en.wikipedia.org/wiki/Russian_web_brigades