OK, let's get a couple of things straight before the gunfire starts.
1. I don't ski bumps as well as I want to.
2. I do OK, by the standards of most, although I never look particularly quick or exciting.
3. I've skied a few bumps, mostly during years spent at WP/MJ.
4. I only have one pair of boots. A pair of Nordica plugs. 130. No shin bang.
5. I only have two pairs of skis. Neither is a dedicated bump ski or racing ski.
6. I will be 60 entirely too soon, but I still ski what passes for bumps at Whitewater and my knees don't hurt. Sometimes I still get to visit MJ. My knees don't hurt there, either.
7. I also ski powder. Lots of it. Often with unseen lumps underneath.
Sooo, FWIW, IMHO, Your Mileage May Vary, etc., etc. but...BWPA is right. You get leg burn and knee pain because of technique. Period.
In particular, even though you don't have to keep your hips over your feet all the time, they do need to be over your feet at the right times. Your legs hurt, your knees hurt, because you are using muscle (since you work out, you have a lot of muscle) to keep yourself upright. You don't have this problem on groomed snow or the race course because you are more confident there and you are more accurately balanced, though not ideal. The imbalance is still there, but you don't have to work so hard to prevent yourself from sitting down on smoother snow.
You don't like bumps. You sit back a little. The uphill sides knock you backwards a little more (as is Right and Proper in bumps) but you don't catch up to your feet going over the top. Your survival instincts keep you back a little - it doesn't take much - on the downhill side or face, your feet accelerate on the downhill side, and you arrive at the bottom, the trough, even farther back. Well, your next turn is gonna be late 'cause from where you're standing, turns are tough, but, hey, you're strong, and you force a turn anyway. Do 6 or 10 or 12 of these, and your legs hurt. No surprise.
So, here's the deal. For racing and for bumps, you have to do a better job of moving with your skis. When you get behind them (and you will get behind them), you need to move both forward and laterally into the next turn. Too much of either, and the turn won't work as well and will take more muscle. For racing, this will give you a cleaner turn and better times. For bumps, this gives you accurate control from the very top of each turn so that you can scrub off speed with little effort, if you want, and use less effort trying to catch your feet and keep your butt off the snow.
In racing, running the gates is the fun part. But if you're serious about getting better, you have to drill and drill and drill some more to learn exactly what to do with those tools on your feet to get the fastest, cleanest run. It takes discipline. If you're not doing the homework, if you're not getting coached by someone who knows what they're doing, then you're not serious about racing, and it's just something you do to reduce boredom.
For bumps, well, running the bumps is the fun part. Like racing, it's challenging. It's different - it's a continuous challenge to your balance, for one thing, since the slope changes from flat to up to flat to down (sometimes very steeply down) to flat again all within a few feet. Still, the fundamental skill set is the same for both. Some of the drills will be the same for both. The duration, intensity, rate and timing are different, but maybe not by as much as you think. And guess what? It takes discipline, drills and practice. Some of the drills are even the same. And most good racers ski bumps quite well. Racing skills feed bump skills. Bump skills feed racing skills.
Can you control your balance and edging accurately enough to do the simple pivot slip without moving out of a 10' wide corridor? Even though this exercise looks easy, very few people can do it.
Can you take it to the other extreme and execute a series of clean, pure, railed carves where you maintain at least some speed control and avoid the infamous park 'n' ride?
Can you leave a ski at the bottom, mount and dismount the chair lift successfully, and ski - well - on the remaining ski? On either foot?
How many turns can you make between here and, oh, THERE? Not Z turns. Round turns. Can you make short, round turns with enough finish so that you can ski a moderately steep blue groomed slope without gaining speed within a single groomer width? At least one of the keys to both bumps and gates is being able to cleanly ski any line you want no faster than you want. After all, if you're too fast in this gate, you won't make the next one - but you never want to have to just scrub speed. If you can't shape your turns effectively and at will, your times will be longer and the bumps will be harder.
Can you pivot slip in the bumps so that you pivot over the top and slip, slowly, down the other side without trying to "dig in" and put on the brakes? You can use gravity to help you turn rather than fighting it to slow down.
It goes on. There is no one big secret to either racing or bumps. Both require time, discipline, training, coaching. You won't learn either one on the Internet.
Best way to learn to ski bumps? Find someone who knows how and can really teach. Get serious about the drills. Do them. Practice. Ski the same line over and over again. Then move to a different one and ski that.
If you just wanna do a little racing while you're on an expensive vacation in WP? Learn where the bumps are and avoid them. WP has lots of groomers. Enjoy your racing. But you'll be missing some of the best of what you paid for.