I feel that muscular strength does play a role in one's aikido practice, as it plays a role in our everyday lives. However, relying on strength (i.e. muscling through a technique) to perform our techniques can be a hindrance. I've noticed that when you are stronger than your training partner it is much easier just to "plow" right over them without paying too much attention to using proper technique. It is easy to fall into the trap of thinking your technique works when you are stronger. I'm a woman but I'm not a petite woman. But what I've noticed training with smaller woman, and some men (not that many btw), is that I can't feel the technique as well if I'm not paying attention to my partner's energy. A smaller partner gives me less energy and when I'm stronger I try to use that energy versus my own. In order to do this I tend to go slower as I'm looking for connecting with their center, looking for their joints locking, etc. It's not an easy thing to do.
As far as this debate regarding women having to rely more on technique, generally, that is true, but that goes to our body type and the fact that women's bodies do not have the upper body strength of a man of comparable size. Men smaller than me, can easily bear down on their grips much better than I can bear down on theirs. Should you go easier on my because I'm a woman, I hope you only go easier on me because of my skill level and your skill level to find my center and energy. If your only reason to "go easy" on me is because I'm a women, yes, I find that condescending. I just deal with that by asking to be thrown a little harder or by just upping the intensity of my attacks. When I eventually do get thrown the way I like I come up with big smile and with a comment like, "that's the way I like it."
What us smaller and typically more flexible ukes do provide to stronger partners is our ability to stick to you and move with much more agility than our bigger partners. I had one partner comment on how hard it was for him to deal with my flexibility.
Finally, in regards to the weight training question. I also strength train which does aid my ability to do aikido in that it has improved my overall fitness level. It has gotten me in shape so I can train at more intense levels and take breakfalls without feeling like I fell on a pile of bricks. If you incoporate flexibility training with your weights it shouldn't hinder you ability to perform aikido.
Just remember, to not rely on your strength to be effective because eventually you will come across someone who is stronger. It's just us smaller people come across that a lot more, and are forced to focus on technique rather than just making something "work."