As Freedom of the Hills also states:
"Mechanical ascenders are stronger, safer, faster, and less tiring."
– Freedom of the Hills
The key advantage in there being, "less tiring", yes you could save a couple hundred grams by not carrying ascenders and instead using some prusiks, but how much do those weight savings benefit you after you've ascended a rope a few times fighting with tight knots and all that extra friction?
Mechanical ascenders should be used in any situation where you plan on using a static rope as your main mode of ascent versus carrying dynamic rope as fall protection. If you're free climbing, then it's not necessary to carry ascenders, and prusiks will suffice if a situation arrises where you need to ascend the rope.
You very well could use a prusik in every situation you'd use an ascender, but ascenders were invented to replace prusiks, their only disadvantage is their superior weight, and if you're super picky you could maybe also complain about their superior size. But the time and effort you save on an ascent while using ascenders more than makes up for the effort you put into carrying the extra weight.
Your other suspicions are spot on, prusiks can be absolutely useless on iced-up static ropes, I've been in icy situations where my prusssic wouldn't bite the rope at all. Mechanical ascenders will never slip (unless it's completely worn out and doesn't have anymore teeth to bite with).
In aid climbing it's a no brainer, if you're laying siege to a big wall, you'd run the risk of wearing through your prusiks from the milage you put on them, also you're not too concerned about a tiny bit of extra weight because you're not climbing with all your gear, you're hauling most of it. Ascenders are what you use to haul your gear anyways, nobody is going to choose a prusik over an ascender for that job.
For mountaineering, fixed lines are used for high traffic routes on slippery slopes, typically if you're climbing a route with fixed lines, then your ascender is likely going to be the only piece of gear in your rack, the only other thing you'd be carrying would be an ax (unless you're the guy fixing the ropes). I'll use Everest as an example: Everest basically has a fixed rope from the bottom of the Lhotse face all the way up to the summit. Congestion is a problem as it is, I can only imagine how infuriating it would be for a climber queued in line behind a guy using a prussic, "Hold on just a moment while I tie my prusik onto the next rope..." Aside from the increased risk of frostbite from all the finger work, the extra time spent fiddling with your knots could put you at greater risk of getting caught on the mountain after dark, not to mention the fact that prusik knots will not grab the rope unless they are tight and weighted, if you slipped while following a fixed line with a prusik, there's no guarantee that the knot would catch you, at least not right away.
If being weight conscious is your only hang up for carrying ascenders, then I'd recommend getting couple of these:
I never go anywhere without one, it's got all the biting power of an ascender at only 39g. I still recommend carrying prusiks, but they should be used mostly as backups.
One advantage that is noteworthy for prusiks, is that you can also descend a rope with prusiks as easily as you can ascend it. Ascenders are one-way, so you have to stick your fingers in the device to get them to release.