I've been ignoring snails for the most part. My catfish will eat them when they can.
Only real problem I've had with the snails in my system is that their little shells can clog small holes very effectively. Like spray bar holes or the holes feeding my towers or the little drain cup hole in the indexing valves but other than those issues, they seem to be mostly eating detritus and not bothering my plants so I'm not stressing about them. Been in my systems from the beginning pretty much back in 2008.
I have devised a trap for the snails that will let me mange their population. Crush 'em and feed them to the fish. I got a tip from an aquarium site to blanch some lettuce and stuff it into a jar with a small lid +- 1". I managed to catch 10 or so over night. I also read that clown loaches will eat them.
I live in a very damp and cold area, so far very few snails, but many, many large slugs. I haven't built my greenhouse yet, but I am planning on carefully incorporating strips of copper into the design to try to prevent slugs from getting started. I may also use the greenhouse as a nighttime shelter for ducks. I have used loaches in my aquarium to try to control snails with little success.
The trick with the copper is to make sure it is just a barrier to keep them out of the grow beds and avoid having the copper in contact with fish water since too much copper in the water would be bad.
Try things and see what works.
Yes, my thought is to put the copper around the perimeter of the greenhouse, as far from the fish as possible. A few beer traps will not cut it. There's a swamp out back of my house and without serious efforts at control, I have major crop losses in my dirt garden. I doubt it will be different in the greenhouse.
Re the disease issue: Yes, slugs and, I believe, snails, carry bacteria, including staph and e. coli, which are deposited in their slime trails. Regular soap and water are generally ineffective when cleaning slug slime off of contaminated produce. I usually spray with vinegar before a regular produce wash since vinegar kills some germs and I think it helps to breakdown the slime. Virtually all my ground crops and 30 to 50% of my trellised crops have slug damage or slime.
Well I don't dare let my chickens and ducks into certain parts of the garden since they would eat all the plants but I often let the birds roam the whole yard except for the netted off garden beds so the pests have to get through the gauntlet of birds to get to the garden beds.