The SS (lol, the irony) are generally just as retarded as the regular FBI, and are biased in the same ways the FBI is. I expect the only reason they supposedly liked working protection duty for Trump was that he treated them very well, unlike (apparently) most politicians who live and/or work in the White House who tend to treat them like shit.
They absolutely would pay an idiot a visit for making some off-hand remark that no one would take seriously, even things that are deliberately and quite clearly a joke. SNL once did a bit where several performers talked about how hyper-sensitive the SS is about it and each of them said things like "yeah, you really can't get away with saying [looks directly into the camera] 'I'm going to k_ll the president' or you can get in real trouble." And yes, the SS did give them a stern talking to for that.
They're bored, stupid and on a power trip. They're also ostensibly trying to protect the president from genuine threats, but since there really aren't any, they have to try to look like they're doing something to justify their absurd budget. And I guess getting mugged while escorting the president doesn't count as "looking busy."