I personally found very hard already to progress on weightlifting during diet cycles, as I need to cut on volume significantly; can't imagine to do that while fasting, unless fasting would "flip a switch" in how the (my) body works, which I doubt.
Allow me to step right up.
First, as you might expect, aerobic and endurance exercise benefits greatly from being in a fasted (16-24 hr) state. You're light, quick and your body is devoting zero blood flow (and other resources) to digestion. I suspect you're not surprised by this.
Second, I do a 1.5 hour heavy leg workout which includes squats, deadlifts and other assorted "big" movements with weight that is significant, compared to my body weight. No issues at all. I wouldn't do this on a 24 hour fast, but it is my regular habit to do this workout while in a 16-18 hour fasted state and in general I have very slightly better results than if I had eaten recently. Again, the issue is the amount of blood flow (and other resources) your body is putting into digestion.
Finally, while it is easy and enjoyable to do my BJJ workouts in a fasted state, I cannot, then, do several rounds of sparring at the end without more recent calories. Or rather, I can but I after round one or two I am running out of gas.
YMMV. You should just try it. It costs nothing.
If that's true, why isn't it a universal practice in amateur and professional athletic competition?
(Or, for that matter, why isn't it banned?)
When you are right at the edge of your abilities being just a couple pounds lighter can feel like a world of difference (people hone down their rack to precisely what they need for a route, nothing more), especially on sport and boulder problems. My anecdata points to it being great for the project/performative aspects of the sport but as a tool for consistent progression/improvement, I am less confident in stating anything.
I'd say it depends greatly on the effort and the individual. Long, slow distance for someone who is accustomed to training fasted? Fat stores will suffice as an energy source.
But add in some intensity and that's gonna be rough without much glycogen in the tank - which will be the case if you've been fasting for the past 16-24 hours.
First, fasting does not require that your overall calorie intake is reduced. If your cut-goal is 2000kcal per day, you can fast in the morning and eat all 2000kcal in the afternoon. Ultimately, I find this is actually an easier and less hunger-inducing way to each your cutting goals.
Second, you're right. Doing weightlifting while fasted is hard. It's better to coordinate your gym schedule with your eating schedule so that you can eat -- I like a simple protein shake -- before and after the gym.
Third, I'm seeing a lot of folks talking about intermittent fasting ("IF") and day-long fasting in the same breath. I think IF is great for weightlifters on a cut diet. But I think day-long or multi-day fasts are different and should only be done occasionally (weekly being the most frequent) if you are trying to progress with weightlifting. YMMV.
The studies may not have been conducted among athletes so, testing your own body's response (by measuring lean body mass before and after a fast) seems like the best way forward.
I have read that people on keto diets or fasting have less "explosive" energy for workouts. I am not concerned that my body is going to cannibalise my biceps if I don't eat for a day or two. There is a lot more energy kicking around that is less important.
For me, it's worth it for the HGH boost: https://www.dietdoctor.com/fasting-and-growth-hormone