IMO it might be possible to go up Rainier in warm clothes and kahtoola's, but I wouldn't do it --- it is indeed an intense experience. To put this in context, I plan to thru-hike the PCT this year in trail runners, including the Sierras and climbing Mt. Whitney. And no overshoes, I'll just use goretex socks.
IMO for Rainier you don't want a lot of weight compromises on gear, for example, I wanted excellent crampons, and of course boots that work with those.
You said: "we do not have an interest if it requires technical mountaineering caution". I think then that you do not have an interest ...
It is a technical climb, and I agree with the majority opinion that you need to wear a helmet and be part of a rope team. An experienced rope team, with someone leading it who knows the route and is willing to turn back when the whether dictates that.
There's a substantial difference between different mountains in terms of what's needed to summit with what most would consider a reasonable degree of safety. The death toll from unprepared climbers going up Mt. Rainier is a whole lot higher than for those going up Mt. Adams, which is the second highest mountain in the state. What you propose could work for Adams (I'd still be inclined to go with more, but it could work). And BTW, Adams is a fun climb, I have to say that while I'm glad I climbed Rainier, I *enjoyed* the climb up and down Adams a lot more !