If there are so many other ways to quickly and effectively clear the crease as you say there are, then why does so much crosschecking occur there?
Because it's allowed. Why would you NOT take that extra leverage on your opponent, and the opportunity to inflict pain and maybe injury on him? If the ref's going to let you do it, of course you will. I know I do when I play. It's a conscious choice to NOT do something legal, because I know I can get away with the illegal move to get a better advantage.
Simply leaning on a 6' plus over 200 lb player is not going to get the job done so as long as a certain degree of crosschecking is tolerated, it will continue.
If you establish position correctly, there's no need to move him because you're already occupying the spot he's trying to get to. It's up to him to figure out how to get you out of the way. If you use your lower and upper body properly, you can manipulate him as he tries to get past you. If you have your stick positioned correctly, you can deny him the opportunity for tips or rebounds. This stuff happens constantly in any game, using effective techniques that are coached from the very lowest levels on up. If a defenseman can't figure it out, why should he be allowed to illegally "punish" a forward who is outplaying him?
I'm not saying crosschecking is the game's biggest front-burner issue or anything, but it's one of those things where there really isn't a rational explanation for why we all just agree to ignore the rules and let defenders take illegal advantages. It reminds me of the late 90s and early 00s where everyone just decided that hooking and holding didn't need to be called anymore. I distinctly remember the argument that physical contact was just part of the game, and defenders "needed" those techniques to do their jobs. Then we took a year-long break, eliminated that aspect of the game completely, and lo and behold it turned out that those illegal techniques were actually completely unnecessary all along.