The funny thing is even in that sort of hostage scenario, they’re much better off sending Jack Slater in instead of a SWAT team.

Who's that? Obviously the Hostage Rescue Team exists, but most hostage situations don't last for days and getting a specific group of people across the country to the scene is harder than getting a team there that's already in the city.

Are they really sending in the actual SWAT, or are the regular cops just now so well armed and body armored up they look like SWAT to civies?

What's the difference?

a whole lot of difference, and it's sad you even have to ask. actual cops dressed up don't actually have the training SWAT does. everyone forgets the And Tactics part of the acronym and just focuses on special weapons part, but street cops aren't going to have those tactics. they're just going to have big gun with armor and giant egos and possibly scared shitless and prone to make mistakes.

The problem is that every rural county sheriff's office and podunk city police force now has a "SWAT" team that has nowhere near the training that the early examples of the idea have.

That Sheriff's office not only has a poorly trained SWAT, they also have a military surplus armored personnel carrier, a few shoulder launched rockets just in case things get a little dicey, and extensive reinforcement of the idea that everyone's out to get them and they have to react quickly with the maximum amount of force or their lives are in jeopardy.