The only weights that I use these days are two lb. weighted gloves for shadowboxing.
I'll probably get a weighted vest at some point also for certain movements.
Bodyweight movements will build an impressive physique. One look at a gymnast will confirm that, and, as you pointed out, they involve the core heavily, and core strength is a must have for anyone involved in a combat sport.
You can use bodyweight movements to build max. strength, strength endurance, and explosive strength. Lots of possibilities.
My favorite thing about bodyweight exercises is that you don't need any equipment aside from a pullup bar and perhaps a weighted vest, and you don't need to pay for a gym membership for the benefit of waiting for some sweaty ape to finish up on the bench that you want to use.
There's a certain coolness about building a great physique using only the weight of your body.
Insert profound martial arts quotes or tough guy phrases here.