I always supplemented my martial arts practice with weight training. The two belong to each other, for me personally.
When it comes to fighting prowess, there's no question that well-trained muscles will give you an edge. And it's not just about hitting harder... You will move differently overall, having a fair amount of weight training under your belt.
However, raw strength is no substitute for sloppy technique! In many cases, improving your body mechanics will give you a greater gain in power than any amount of resistance training would. That could actually be said to be the essence of martial arts training on the physical level: Learning how to use what strength you have to maximum effect.
As far as self-defence is concerned, there is yet more to consider. Functional SD involves exploiting your adversary's weaknesses. If you are thinking of it as setting your strength against theirs, you are on the wrong track, in my view. How are you going to teach women to effectively defend themselves against men twice their size and four times their physical strength on such premises? You've got to emphasize evasiveness and hitting vital targets here... That kind of thing. And this holds true for functional self-defence in general.
That being said, having more strenght equals having more options when in a critical situation. For example, holding your adversary down rather than knocking them out. Or stunning them by a punch to the solar plexus rather than crushing their throat. More power to you!
At any rate, fitness matters, and it doesn't include only strength. Flexibility has been mentioned as important in this thread, and I fully agree. This is not just about being able to kick people in the head, even though it's nice to have that skill.
Again, it's about being able to move smoothely and swiftly overall, and to keep this ability in the long run.
If anything, weight training and stretching become more important as you grow older. But be reasonable and don't overdo them. And give your body enough time to recover. You want your exercise to be supportive of, not detrimental to your health.
While I was all about heavy lifting in my younger days, now that I am in my 50's, I prefer a more mixed approach. Kettlebells are great. Wrist and ankle weights as well as resistance bands are useful too. Add on some good old body weight exercises for good measure.
I really enjoy experimenting with different kinds of equipment and training methods, and seeing in what way they help improve my power specifically. Giving your body a new type of challenge will keep it on the ball.
Often, I will make those different kinds of exercises part of a Tabata routine. Tabata and related forms of HIIT are excellent ways of exercising power, stamina, and flexibility - even all at the same time!