There is no substitute for moving your body's weight through space.
I know that when I have a healthy balance of weight training and bodyweight training, I feel the most badass. I really don't care what your 1-rep bench press is is, or what trophy you may have won in some non-contact martial art tournament. A healthy balance of all facets of fitness and fighting conditioning is what matters if you meet me in a dark alley. Bodyweight training is very, very important in building "fighting strength."
At the same time, if you only train bodyweight, you sacrifice maximal strength and the ability to move external loads, and will never move your strength to the next level. Take two martial artists of the same size and skill. The one with the most maximal strength will dominate. It carries over into everything.
Here are a few things I have put together as a measure of fighting conditioning:
6 rounds of...
6 barbell deadlifts @ 405lbs, followed immediately by a 440yd run.
These were continuous rounds: Come in immediately off the 1/4 mile run and lift 405lbs 6 times.
Repeat 6 times. No rest. No whining about how heavy the weight is. Can't lift it, or are too gassed from your 1/4-mile run? Too bad. Your enemy will not care about your excuses or lack of preparation.
500 bodyweight squats
100 ring dips
No strength endurance? No strength to pull or push your own bodyweight around for 700 reps? Too bad.
Last week, I threw this out as a diagnostic:
4 continuous rounds of:
5 deadlifts @ 150% bodyweight
5 tire box jumps at at least 50% body height
25 ring pushups
No rest. This was a measure of strength-endurance, power production, physical agility, strength-endurance, relative body strength and cardiovascular efficiency.
All of these were mental toughness workouts, which take confidence and discipline. These are very important attributes, with regards to physical combat.