Speaking personally…
I think it makes more sense for the fretting hand than it does for the picking hand. I’ll I guess add the caveat that I’m not sure this is as true for coordination related stuff as it is for, say, legato patterns… But a legato run is mechanically the same at 16ths at 80bpm as it is at 160. That’s not really the case with your picking hand, where your picking mechanic IS different at lower speeds than it is at higher ones - the whole challenge of changing strings is essentially a non-issue at lower tempos since you can just lift the pick up away from the strings in a mechanically inefficient manner and not really pay for that.
Eh, I don’t really agree. The “problem” with changing strings with fast alternate picking is getting the pick up and over a string while moving to the next one. That doesn’t really exist because at slow speeds almost anything works since losing time for a mechanically inefficient motion isn’t really a concern. You pretty much HAVE to play at tempo before the efficiency in which you’re getting the pick over that obstacle starts to matter. It may be possible to learn something by doing it slowly before you do it quickly, but we’re not talking about terribly complex motions here that you should be able to get up to at least a decent speed pretty much as soon as you realize what they are… and, you’re not really going to have any way to tell if what you’re doing is right until you ARE doing it rapidly.
I think there are definitely things where it makes sense to start slow and build up. I just don’t think the physical process of moving a pick through the strings is one of them. Or, at a minimum, if a pickstroke IS something where there’s an upside to starting slow and gradually going faster, then I’m going to require a pretty compelling argument.

