This. Exercises are tools in a toolbox. You can't use the wrong tool for a certain job and then say "See! I told you that wouldn't work." Ring dips and weighted chin-ups are killer exercises for grapplers, but so are push presses and turkish get-ups and farmer's walks.
"Noticeable" is a vague term, but yes it's possible to build muscle (up to a point) using only a weight vest, or only one kettlebell or only a barbell. The biggest hurdle you'll eventually hit is finding a way to progress in volume and/or load (possibly tempo) before having to change to a more challenging exercise variation.
Sounds like you're almost talking about "functional training", which has become something of a dirty phrase because it's misunderstood and misapplied. I'll just say that variables like volume, frequency, and general exercise selection are more important in that regard, rather than thinking in terms of "just bodyweight" vs "just free weights".
It's just as easy, maybe easier, to come up with a crappy bodyweight/weight vest routine compared to coming up with a crappy free weight routine. Focus on the bigger picture - your specific short and long-term goals - and use any/every method available to reach those goals.
I disagree. A bodyweight power clean, for example, will "work" the body more completely than pretty much any weighted bodyweight exercise I can think of. It will also go more towards building the strength, speed, and athleticism a fighter needs more than weighted push-ups, pistols, or gymnastic holds.
Then your programming (volume, intensity, and/or frequency) is wrong. That's not an issue with free weights.
Do you mean overhead press or bench press? In either case, a well-designed program would've had you squatting more than "a little more" than what you can press.
Also, your profile lists your weight as 170 and your height as 175. Is that pounds for weight and cm for height?