Also, on choppy/hard/steep crud off piste, I find it hard to hold a good turn on the Toeside while absorbing the bumps.
Take my advice with a grain of salt as I haven't ridden tight trees a whole lot. But when I've been trying to crank tight turns in a race course, I've noticed that smacking my face into a gate is no fun. How you make your toe side turns could have a lot to do with it. What I've seen effective in tight conditions is a heavier reliance on ankle movements so that you can make turns with your upper body more vertical than normal. These types of turns don't allow for a lot of lateral movement across the fall line, but they do allow "snaking" through obstacles like trees. Practice this on cat tracks - alternate standing on toes/heels only - no steering of the board and no lateral movement of the upper body.
On steeps the trick is dynamic carving. The hips get lower to the snow so the legs can get further out from underneath the body and the board gets on higher edge angles. Higher edge angles cut through the crud better. Also make sure you arch your back a little on toe sides.
You certainly can turn in soft snow by pushing the tail. I see it all the time in the pro videos. It's fun and it works. But that technique just won't work real fast in tight spaces. The board is going to maneuver a lot more nimbly when the pivot point is in the middle of the board vs the nose.
Without seeing a vid, let me make some assumptions that I had to over come myself since your problem sounds familiar to me -- however it may not apply.
Since you can "crank" the board around heelside, and are having trouble toeside, I am betting that you're *overextended* when you make that heel turn ... that is, your front leg is straight, and your back collapsed with the board far ahead of you when you finish your heel turn. It really slows the turn initiation to your toe since you have to make a big move back to the center of the board to start the turn ... time you may not have as you are inside the trees.
Rusty's drills are really good (hips low, flex the ankle especially the front to start the turn, pivot comes from the middle not the front); however, I might add that none of these will be successful if your alignment is off when you start the turn in this locked position.
That you say you have trouble in bumps and crud with the turn makes me think my guess is on the right track. Being out of an athletic stance gets exposed dramatically on uneven terrain.
You know when you are locked on regular terrain, when you can make short, snappy "scarved" turns of consistent shape heel and toe. If my diagnosis is right, start pushing the board away from you rather than around you earlier in the turn, and when it bites collapse into a neutral stance before starting the new toe turn.
See Caleb's video segment and some of the same drills and same advice.