Hi Luke,
Sore legs after Level 3 is pretty common, and is an indicator that you were using the right muscles to stabilize yourself. If you had sore arms we'd be worried
I'm certainly no fitness expert, but perhaps if we look at the ideal scenario in terms of stabilising the rider on the bike, and the muscles required to do that, we can figure out some direction in terms of training...
Something that seems consistent when you look at top level guys and their training routines is that there's a lot of cycling incorporated into it. It makes sense right? Riding a push bike puts you in a similar position as a motorbike, and the muscles used are also similar.
Core strength also seems to be a common denominator, and again this checks out. If the legs are strong and can hold onto the bike, and the abs and core are strong and can control the movement of the upper body mass, there is less chance of the rider using the handlebars to stabilize himself.
Something I read in a Health and Fitness book written by a couple of scientists was really interesting. They said that if you have enough strength to perform 1 repetition of any movement, then the training should be geared towards endurance, as opposed to more strength. Typically, I thought about that in riding terms, and figure if you have enough strength to perform all the necessary actions on a bike at least once, then there is no need for additional strength, only more endurance. They're quite specific in how to go about this, but to simplify it means lots of repititions with low load, as opposed to lifting heavy weights just a few times.
Anyway, maybe that gives some direction for you.