So that’s yes to investigating to determine if threat exists or not, right? Otherwise, how does your quip make sense? Who determines if a “threat” is just “aggressive” words, whatever that means, or a true threat?
Ultimately the courts decide. But you do a "sniff test" of the complaint before you investigate anything at all. If it's obviously not going to meet any legal threshold, stop there. If not, then next talk to local law enforcement that punted on it, do another sniff test at that level, and so forth.
The important thing is more intangible, which is the disposition of the feds. If they're on a "mission," then that could go in a scary direction easily, as post 9/11 taught us. The disposition should only be "we're the last resort if local authorities egregiously fail to act. We aren't going to intercede in borderline cases." Abstracting away the current issue, I'm wary of expanding the venn diagram of acts that can prosecuted both at the state and federal level, as giving government two bites at the apple.
Out of curiosity, would you support federal prosecution of rioters and looters in, say, BLM protests, when state and local authorities fail to act?