uncaged tamed spiders still wave nets, but They die pretty fast (one year), so you just waiting your dwarf life.
...and you need to replace them, and again... too much work for dwarves and too much bother micromanaging this. Especially compared to webslingers silk farm. Then again, when
that stopped DF players?
The good side is: constant exercise for your trainers, preparing them for work on more valuable critters.
And since Animal trainer is one of few skills that involves
knowing the beast Intuition (other than Appraiser and Judge of intent), training critters all the time may help to keep your broker sharp while doing something useful, too.
You need just have closed some mossy area, where your cats will have no enter and spiders will just spawn eventualy.
...oh, and yes, if they spawn on your map, there
also will be wild spiders still spawning - which in itself is a good thing, as you need replacement, but dorfs will go and collect their webs along with that of domesticated spiders and get bitten
anyway. Unless you limit web collectors to burrows that don't contain areas where wild spiders may spawn.
So, can be done, but entails more micromanagement.
Bites are causing negative though, but not too bad one.
But cave spider bite causes mild, but
permanent syndrome and brown recluse spider bite causes necrosis... phantom spider causes mild temporary syndrome - but it lives in evil forests, which is already bad enough.