The camp's that don't believe in weights, have their opinions based under the premise of not wanting to add movement binding muscle. However, that premise is a fallacy because there's a misconception in the correlation between muscle hypertrophy, muscle strength and muscle power. Old timers, ESPECIALLY the Asian culture, feared being "muscle bound" by too much muscle, so weight training was taboo and all your strength training were primarily BW.
However, you can be stronger and more powerful [those are two separate concepts], then someone who possesses more muscle then you. Muscle size, is just the hypertrophy of the muscle cells. Strength, is the mechanical efficiency of the muscle, ie. muscle fiber recruitment by motor units per contraction. Power consists not only of superior muscle fiber recruitment by motor units per contraction, but also coordinated motor unit firing, increased neuron firing (of acetylcholine) and decreased excitation of the golgi tendon and decreased co-contraction of antagonist muscles in high velocity situations, such as a strike.
Without delving into the details of all that mumbo jumbo, YES, weight training is beneficial and a superior form of training as oppose to exclusively BW. Yes you can train core muscles without weights, but nothing you do with your body weight (such as a plank or weighted crunches) will equate to the kind of core stimulation dead lifts or front squats incite.
Yes you can train power using body weight, plyometrics are primarily body weight exercises, however, plyometrics train the stretch shortening cycle, which may help with an explosive TD, but it's not training your hip extension [& flexion] which is the absolute core power of your strikes, TD, TDD's and sweeps, like doing cleans and snatches.
I'll try to make some analogies to create some clarity in all this mumbo jumbo.
Here is an example of muscle efficiency, destroying superior muscle mass/volume. If the amount of quads, glutes, hams, calves muscle determined how fast you can run, then pro bodybuilders would be the fastest people on earth, but they are not at all. Olympic sprinters, though they are muscular themselves, they are dainty compared to a Ronnie Coleman, but the sprinter possesses explosiveness, which is the amount of peak force you can generate in a single moment, therefore, with each stride, the Olympic sprinter is outputting Zx'Z amount of force against the ground in a split second. The pro bodybuilder, though he can out squat the sprinter by hundreds & hundreds of pounds, his muscles aren't primed for power, which is the ability to generate peak force output in a single burst, so with each stride, his output is only Z (as opposed to Zx's Z). Now if the bodybuilder had 3-5 seconds to generate that force, the he'd be dominant, but that need for the extended duration to produce max force is an example of strength, whereas, power (like strikes, kicks, TD's, TDD, Sprawls) are a display of power, quick, maximal single bursts.
Core strength or strength period, is displayed as a Brock vs Mir II. Power, was Nate vs Maia. You can be strong and slow (aka lacking power), you can also be the smaller, weaker guy, but posses superior power.
Size, Strength and Power do have correlations, but at their root, they are three separate concepts at the upper echelon of each attribute. You don't need to be muscular to be powerful and core strength only includes your abdominals, hip, pelvic, lumbar region, which does not require you to be swole to posses.
Use weights was the gist of my argument.