You've made a leap of assumption that all architectural paint is water-based. That's wrong, but if you want that, it's perfectly possible to paint a house using entirely water-based products. You just have to avoid higher-performance products like Kilz Original.
There are different kinds of paint. You need to read the labels.
The paint label will specify what to use for a reducer (aka "thinner" that makes the paint thinner). Whatever that "thinner" is, you can use that for cleanup solvent. Some paints will recommend another chemical for cleanup if the reducer is too expensive.
Now, Kilz Original is an alkyd (read: oil-based) paint. It says "Do Not Thin", because if you reduced it with paint thinner, the paint-in-pot would exceed the allowable VOC limits for some regional Air Quality Management District somewhere, and they don't know where you'll paint it. VOCs are mineral solvents like gasoline that evaporate and cause regional smog. Los Angeles being the poster child for this; their rules are far more draconian than everywhere else.
The Kilz label indicates using mineral spirits for cleanup; that covers a wide variety of products including paint thinner, literal mineral spirits including odorless, etc. In the halcyon days of my youth before anyone knew better, we used to use 60 cent a gallon gasoline, of course disposal was an environmental nightmare even then. Nobody wanted bare spots on their lawns.
I'm pretty good at cleaning brushes with just a couple of ounces of thinner. I paint stuff I don't want til the brush goes dry, re-whet the brush in contaminated thinner, work that in, paint stuff again til the brush goes dry, re-whet, then use rags to soak the thinned paint out of the brush.
If you have been using water to thin alkyd paint, that would only make it thicker still - and spoil it too!
It really helps to use the right brush, though.
"Normal" hardware-store rollers can be used with alkyd paints/primers like Kilz Original. However "normal" synthetic brushes will not perform well with alkyd paints -- with alkyds, use natural-bristle brushes. Conversely, do not use bristle brushes with waterborne paints -- they will water-log and perform badly.
Bristle brushes are readily available, in any quality from cheap Indonesian chip brushes to badger brushes like your great great grandfather used. I use the cheapie brushes by the 24-36 box, because the LPU paint I like has very expensive solvents, and I'm not spending $10 in solvent to save a $8 brush lol. So the idea of lovingly caring for an expensive brush is pretty alien to me, but in the 20th century many painters did that their whole careers. All paints were oil-based then.