I have seen no evidence that they're better than the alternatives listed above.

They are better at tasting minty though! ;)

typically sodium laureth sulfate. They've all also contained some sort of fine particles of some calcium-containing substance, but I assign no expected utility to that.

I feel a significant difference when I use a paste containing triclosan than one just containing the detergent - in particular several hours later when using the triclosan-lacking paste I'll get the "bacteria is building up" feeling/taste/sense. I have not compared this to peroxide use.