It really depends on how deep you want to get into it. The basic composite/S-Video capture devices on Amazon/eBay/AliExpress do work if you want something really cheap. Combine that with a distribution amp so you can play on a TV while you capture. Or if you want a higher-end setup, a RetroTINK-5X plus a component video cable and an HDMI capture device will get you the best possible quality from original hardware.
There are posts every single day about cheap adapters that people buy and then wonder why it looks terrible or there is lag.
Stop buying cheap adapters expecting them to work or be great. None of them will be.
You said you have quite a budget, so use some of that budget and get a RetroTink 2X Pro or the 5X if you really want to take crazy advantage of your 4K TVs.
For PSX, getting a good S-Video cable would be a good pairing and plug that into RetroTink. Plug in the HDMI from the RetroTink to your TV, and you're set.
For PS2, get an HD RetroVision component cable. Plug that into the RetroTink, and you're set.
2X-Pro: https://www.retrotink.com/product-page/retrotink-2x-pro
5X-Pro: https://www.retrotink.com/product-page/5x-pro
PS1/PS2 S-Video: https://retro-access.com/collections/playstation/products/sony-playstation-rgb-scart-lead-ps1-ps2-grounded-cable-cord-lead
PS2 Component: https://www.amazon.com/HD-Retrovision-PlayStation-Premium-Component/dp/B07H3F94ZN
For SNES, you can get an HD Retrovision component cable as well and it’ll look and work amazing with the RetroTink, as the SNES supports RGB.
You can get a different S-Video cable for the PS1, but I highly recommend the Retro-Access cables. If you're wanting the best cable, that's probably the one you want to make an order for and wait until January to get it shipped out.
For the 2X/5X, you should just YouTube the differences and see which version you'd rather have. the 2X-Pro is cheaper and probably going to be just fine for you, but the 5X has some awesome advantages.
People may recommend cheaper adapters like Pound cables, but those are pretty much junk and only recommend by people who don't use better devices or scalers, and are happy with decent at best quality. If you want it to be good with your screens you have now, the most affordable and best option is probably going to be a RetroTink route.
Get one of these instead. It's both cheaper and higher quality than the Sony cable.
Would you recommend just finding any old MVX switch on Ebay for however cheap I can find one? Or is that something I would really want to make sure is in the best shape I can find it?
Extron stuff is pretty sturdy. It's professional equipment, after all, designed for heavy use in work environments. As long as it hasn't been obviously abused, it should be fine from a functional standpoint. From a quick glance, I'd feel comfortable buying this one, for example. Just depends on how much you care about cosmetics. If you're patient, NOS or barely-used stuff pops up from time to time.
Although, from just a couple of minutes of looking at matrix switches, I feel like they range from like $100 to thousands.
A lot of eBay sellers have no clue how to price this stuff, or even what it is they're selling half the time. I'd say you should be able to get an MVX 84 or 88 (8 inputs and 4 or 8 outputs) in decent shape for less than $100 shipped. There are also 12-input and 16-input versions, though those don't pop up as often and tend to go for more. Also, be careful to look at the pictures and verify that it's actually the model you want, because a lot of the time they're listed incorrectly, e.g. an MVX 44 listed as an MVX 88.
I'll see what it takes to convert the PS2, NES, SNES, Genesis, and N64 into DE-15
I'd just buy the cables from Retro Access. For the NES, you said it's RGB-modded so it'll depend on what connector the modder used for its RGB output.
For the PS2, another option is to use a component cable and a component-to-DE-15 adapter. The MVX will happily switch component video as well, not just RGB. Then you can use another output from the MVX to connect it to the RT5X's component input (using a cable like this for example) instead of the SCART input. The advantage of doing it that way is then you could play PS2 games in 480p/1080i much more easily than having to mess around with sync-on-green.