Speaking strictly static
stances aren't important at all for power generation, you can learn to hit hard on one foot, from a variety of stances and methods of movement.
Stances serve a tactical purpose, for instance things done from Zenkutsu dachi are done that way because whatever you are doing (take the simple example of an forward elbow spike) requires forward momentum and posture.
Neko ashi dachi is a backward flinch motion, shiko dachi is an uprooting and throwing posture, which can actually be found in Judo throws, and older battlefield techniques. Look up ura nage if you want an example.
If you stop looking at the end points of Karate techniques frozen in time, and look at what they are actually meant to do it will make alot more sense to you.
Do you need them, well that depends, if you are doing Karate, the techniques are the way they are for a reason..but the fact is that a ton of Karate dojos don't spend any time on the combative bits, preferring kata-as-show, point sparring, and some uninspired static bunkai drills.
If you don't know that reason, or don't actually know what the techniques are for then they won't hold that much relevance for you, in that case, is it Karate? I don't know, apparently plenty of people think it is.
Take a look at Ian Abernethy or Kris Wilder's stuff if you are having a hard time understanding what stances are for..it's really pretty obvious, but if most of what you do in class is typical ranged Karate sparring you will normally be putting like 2% of what is there to use, and most of the stances and other stuff in your curriculum isn't gonna make sense from that standpoint.
And yes...you can spar, do freeplay, go dynamic with the traditional techniques with some safety modifications.
There really is no panacea for hitting hard, there are a bunch of different ways to hit hard "correctly".
So how you stand and move, yeah of course that affects power generation, when you break stances into different points in time and evaluate them from there though (like kata nad kihon)...that's a thing of looking at tactics.
Traditional stances,IMHO, are to rigid to use exclusively in a self defense situation. By that I mean staying in stance at all times
That is just common sense, holding a static posture for any amount of time in a real situation is usually a bad idea, and quickly gets you beaten. A "fighting stance" is the stuff of fantasy. Stances are not meant to be viewed like that, someone who views them like that (instructor, student, hanshi, grand poobah, whatever) does not get what they are for.
Do you think the people that invented the kata you do thought it was a good idea to stand statically in stances while someone pounded on them? I don't.