I thought I'd follow up to my post in hopes of helping out others. I finally got the GBS-C working as-intended with my PC and Switch. Here is what I used and how it is setup:

HDMI Source Device (Laptop/Switch in my case) > This HDMI to component converter box plugged into the YPbPr component inputs on the GBS-C. Other side of the GBS-C - VGA out port (using this cable) with the component ends plugged directly into my consumer Sony CRT TV. Inside the GBS-C software I enabled Component out over VGA and selected 15khz downscale, everything works perfectly including scaling and re-positioning of the image. The only thing to watch for is the GBS-C must be turned on AFTER the source input or it won't sync properly. I hope this helps someone out in the near future. :)

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.