Predators and prey will always maintain some equilibrium, yes. But you can simply look outside and see that equilibrium typically includes a shit ton of other bugs (or birds, or mice, etc). You can also look at homes with serious infestations to see that this organic control alone doesn't typically keep pest populations in check. Nobody has ever gotten rid of roaches by introducing spiders. Sometimes you do get infestations of spiders though, in which case the spiders are usually eating each other.

Generally, inside your home there are other factors keeping pest populations in control. With a few very notable exceptions, most insects do not actively breed indoors because the climate conditions are not correct, or there is not access to food, or they don't like the vibrations and noise. Most of what spiders in your home catch are bugs which get in by accident and would simply die in one way or another.