That was all brushed, and honestly I didn't even use a super high quality brush, just a throwaway. I made sure to just lay it on thick and not mess with it too much. Since it was a horizontal surface you could lay down a lot of mils, and only look for runs on the edges of the table.
I've done a few projects over various stained/lacquered/etc wood with Polyshades. As long as you scuff it well, I think you'd be OK. The stuff I used it on was solid wood furniture but stuff that wasn't really worth a full strip down, but I wanted basically serviceable again. For all intents and purposes, it works like oil based paint, just translucent. People have issues with it due to the expectation of it working like a stain and somehow penetrating the wood, but it's more just like an oil based paint. Same way of brushing out, catching runs, etc. Like an oil based paint, depending on technique/etc you can leave brush marks. Unfortunately most of my pictures of furniture I did with it show my messy house, so no pics, haha. I did espresso for a color on my other stuff. With Polyshades too, some people don't understand you can only make a piece darker, and how the color comes out is sort of random. Espresso comes out a mostly solid color with 2-3 coats depending on what you're going over, though.
Polyshades does reek, it is still an oil based poly, and my house smelled for a whole week using it indoors for that table. So for kids, maybe not. You'd get more money and likely a better result using a Citristrip or similar and then using water based stains and polys, you could milk the hours to the max. A lot of people like Polyshades for rentals and the like, though.
For durability, I knew one customer of mine who just mixed two colors together and dumped them on an indoor porch hardwood floor and brushed it out themselves. It's held up for about a decade. They have a few tables and chairs finished with it that looked great, too. The husband wasn't bad at furniture, to be honest. Wife slaved him to the max on stuff like that, haha. The only other problem is going over an existing finish is like oil based paints, you can still chip it/etc if you're bumping into it, gouging it with a jean buckle, whatever, and there will be no stain underneath, so it'll just show the old finish color again. On my table it's not an issue as it was down to 100% bare wood, but yeah.
For spraying with straight ammonia, my family is messy, and I'm cheap. It's super strong and super cheap, and cuts all grease down to nothing. I recommend it for paint prep for this reason, though, it really is one of the best things to clean a surface. You can also use it to clean shellac brushes, but it will lift shellac on trim/etc so be careful.