I've been told by several snail keepers that I trust that assassins will kill nerites. If they can't do it alone, then they will do a group kill. The vegetable method coupled with removing eggs as you see them seems like the easiest approach to me. It would just take some time to do it.
A clown loach will thoroughly enjoy eating all the small snails. My problem was solved in 48 hours, and I no longer worry about new snails. My nerite snail is large enough that the loach doesn't bother it. Clown loaches can grow large over time, but you should be fine for a year or so. By then, maybe you'll want a larger tank!
Don't mean to jump on your case, but I have two problems with this, call them pet peeves. The first is you should never buy a fish to "fix" something wrong with your tank. This goes for algae, snails, whatever. There are always underlying causes in those situations. Fix the root of the problem and don't just try and use a fish as a band-aid. You should get a fish because you want it, not because of what you think it can do for you.
Secondly, never ever ever buy a fish that you don't have a proper sized tank for. I mean the tank is set up and ready for fish before purchase. You might plan to upgrade in the near future, but too often life gets in the way. You have to move, lose your job, medical emergency, need the money to fix your car...life. A clown loach is a perfect example. Young ones grow rapidly, are highly sociable so need to be in groups, and active. Young clown loaches should be in at least a 4 foot tank, while adults need something more akin to a 6 foot. That's a vast amount of upgrading from a 20 gallon, just to be rid of snails.