You won't find anything that does this off the shelf at your local pet shop. It's possible to build it if you're an experienced plumber, but the short version is don't bother, just get a big canister filter instead.
First, bear in mind that no amount of water changes will compensate for all the problems of overstocking: it won't increase the amount of swimming room or available territory in the tank, for example. If the water's surface area is too small to efficiently oxygenate the full tank, add an air stone or power head to increase the top-to-bottom mixing. So plumbing your tank into your water line to deal with water quality would be getting ahead of yourself if you have too many animals in the tank and no real water movement.
As to how you'd do that though, you don't want to go straight from tap to tank. Tap water is generally disinfected with chlorine or chloramine, and without chemical treatment this is toxic to fish. On top of that, the temperature of the water also needs to be matched to the temperature in the tank unless you can guarantee that it'll always be within a few degrees of your tank water, even after sitting in the pipes for hours on an extremely hot or cold day.
This means you'd need an external reservoir with more than enough capacity to do your entire water change at once; temperature control and some other life support equipment in the reservoir to keep it tank-ready; a way to pump or drain the reservoir into the main tank; and a way to either drain or collect overflow from the tank as it fills. If you literally want buttons to push, you're bringing in electrical components which makes things even more complicated. All of this would need to be custom-installed in place by someone with plumbing experience, and would basically be as permanent a fixture as a toilet or a hot water heater.
In my opinion, the risk and cost are not worth the benefit. DIYing this kind of project takes significant effort and expense to build: for a small tank like you describe, the parts alone would easily run double or triple the cost of the bare system. It would not reduce the amount of maintenance you need to do by nearly as much as you might hope, because the reservoir and all the additional plumbing will have their own maintenance needs. And most importantly it adds a frightening number of components that can fail, most of which would cause a large flood and/or or total loss of everything in the system when it happens.
If you just want to add more biological filtration, a more reasonable option would be an oversized canister filter or a sump. These are cost-effective, reliable solutions that improve water quality and simplify maintenance. They're also solved problems: canister filters are off-the-shelf purchases, and while most sump designs are a little more advanced, there's no shortage of plans available on the internet and none of them are as high-risk as modifying your house's main water line.