What’s the best, reasonably priced, handheld device I can buy to play GameCube games?

If you have a bluetooth controller and an android device, you could just use Dolphin and not even necessarily have to buy a new device. The controller isn't even technically needed, but trying to play a 3D game on a touch screen? Ew.
- Exactly, the cheapest device is the one you already have!

Probably a Steam deck; I've used mine to play Gamecube games and it's worked fine.
- It’s a good choice, but on the pricier side if you just want GameCube emulation.
  GameCube emulation can be emulated on a lot of lower end devices now that cost much less.
  The Steam Deck is a great device overall, though. I love mine and never regretted getting it.

Specify what "reasonably priced means".
I've recently bought a Retroid Pocket 4 Pro (£120) and would highly recommend it. Emulates up to PS2 and Wii, and amazingly emulates many Switch games. Comes with Android so this open up the option for loads of native Android games with controller. This plays GameCube games well, has great battery life, great portability, WiFi and Bluetooth, halls effect joysticks, nice buttons and d-pad and is a great all-rounder. I neglected my Steam Deck for a couple of months after I got this. It cost £120 on eBay (you might be able to bargain down further if you're patient and keep a look out).
Prices are dropping because the Retroid Pocket 5 is out and is a huge upgrade (this is the console I would recommend if you're looking for a stronger and more future proof option, but it costs about £200). If you're looking to save a few bucks then you could go for the Retroid Pocket 4 (non-pro) or Retroid Pocket 3+.....but I wouldn't recommend those as worthwhile savings.
The other commenter is correct in saying your cheapest existing option will be your current phone with a controller.
The other recommendations for £300+ devices makes no sense to me, but that depends on how far you want to push your budget and if you want the console to do other things.

Steam Deck. I suppose you could get a small emulation handheld but I think the steam deck is a better experience all around for like $200 used.

people are recommending the retroid pocket 5, but i had a flip (1) which is internally equivalent to a 3+ and it ran gamecube perfectly. the flip was discontinued (and for good reason, it's battery had a nasty tendency to go spicy pillow) but you can buy the 4 new cheaper than the 5 and there are a few used 3+s floating around on ebay

I personally have the Ayn Odin 2 which is great. The Retroid Pocket 5 is another android handheld in a similar price range. Both def require a bit of tinkering tho

Your phone probably can already. (If you have an android)
Otherwise. A GameCube! They aren't overpriced anymore and are easily modable