I have actually been planning to do something about draw-to-X for a while now, but here are a few points:
Anything you can't get rid of in the action phase (mostly treasures, junks, and greens) are very bad, since once you draw one, every remaining draw card you play draws 1 less. With almost any in your deck, you WILL draw them, and it WILL hurt your ability to function a lot. First of all this is going to make you want trashing. But this means you usually can't afford to green very long. In turn, you get mega-turn implied to you, especially against someone who isn't going for Provinces with you. Any time your greening is drawn out, it's bad news.
As for the trashing, I don't think you actually need to get rid of absolutely everything - if you're stuck holding Chapel, you're fine (especially since you're going to be able to play it, hooray; this is true for many but not all trashers, but even if you're stuck holding e.g. remake, that's not too terrible, if you have 0 other junks).
You do need a lot of villages, because you tend to want to go Village Village Village Village Terminal Terminal and only then draw, which is especially bad because you have fewer cards to start with, making you need a way higher village density.
It's a myth that villages without draw are better than villages with draw here. All else equal, you prefer the draw still for sure. However, the lack of draw tends to be less bad here than elsewhere, plus the extra bonuses you get for balance (from not having draw) usually offset the lack of draw.
Of course, you still have the problem of needing to get a draw card to start, along with villages and maybe some terminals (though it's ok to draw those later), which is tricky, especially because you don't really want multiple of your draw cards at once. So sifting is quite good to get the right mix, but in general these engines are more finnicky and weaker than their "normal draw" counterparts.
Discard for benefit is nice, but be a bit careful relying too much on it. Hamlet is excellent of course, giving you discard outlet and village and pseudo-sifting.