Is driving in France much more stressful than in the US?
Generally, no, not really. But it depends on where specifically in France. Driving in New York City? Fuggedaboudt! Driving in Paris? Mon dieu, non merci! (I've done it a couple of times, I found it quite stressful). Normandy, the Loire Valley - a drivers paradise. In general, driving in the old historic centers of almost any city (even small cities) can be challenging - the streets were narrow back in the middle ages, and still are, but now they're crowded too (and often illegal to drive through).
I thought having a car in the Alsace villages would be generally more
flexible. Sounds like I might mistaken.
Again, that depends. Rural areas can be an absolute delight to drive around. And getting between towns should be no problem, the issues are when you reach your (popular) destination. But if you're in the wine country, especially during "wine festival" season, expect it may (probably will) be crowded in places. And besides, you like wine? When you start sampling any, pretty quickly you're done driving for the day. Along the Route du Vins (around Colmar, just south of Strasbourg), I figure I'll let someone else do the driving, and I'll sample the local produce.
Are there a lot of travelers using the system with suitcases in tow?
Well, suitcases...big, hard-sided, classic Samsonite monoliths? I think most people have figured out that smaller, more forgiving containers make for easier travels. But yes, of course, you'll see plenty of tourists (French tourists and international visitors) on the trains with some sorts of bags. You'll see people moving smoothly and comfortably with their small, light bags, and you'll see others who are very obviously still in the process of learning the advantages of traveling light as they struggle with big, heavy, awkward steamer trunks...
Car rentals are neither inherently good nor bad. It mostly depends on where you are trying to go, and other variables. Some places are easier/harder, more/less efficient if you have your own car (for example: I think Normandy, or western Ireland are easier/more efficient by car, because the public transit systems there are not efficient for tourists; along the Rhine, or in Switzerland and much of Italy, trains are perfectly fine, easy, efficient). There are always exceptions. Personally, I pick and choose the best ways to get around based on my needs, preferences and what's available, often combining both car and rail on the same trip. To connect the places you mentioned (Amsterdam, Basel, Strasbourg, Lucerne) I'd take advantage of the fine rail systems in all those places. The only places mentioned where I would look elsewhere would the "wine villages" and in that case, those are popular enough that there are dedicated services set up to take people (people who have been drinking...) around them.
The rail system does look great.
It is - in some places. One quickly comes to appreciate that! In those places where it's not, there are other options.