GoPro is the go-to for a reason, and they're uncomfortably tiny when not in a case.
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Do you have a recommendation for a particular model? A brief search pulled up only one model just a little above $200. The rest are $300+.
Also, if you’re trying to beat a smartphone optically, modern mirrorless cameras are good. And not embarrassing to hold. This is my little R50V next to an older body:

And I am not a vlogger. It’s my travel/family cam.
Though TBH I’d recommend a used Panasonic Lumix S9 instead. That will utterly blow a smartphone out of the water.
…But it’s not cheap. A used S9 with a 18-40mm lens is $1000, and that’s a steal in the photography world. The R50V with a kit lens is like $700 new, and that’s as cheap as these video-centric cameras get.
Those aren't exactly small and discreet, and like you said, they're way above my budget.
What’s wrong with a smartphone?
I don’t mean to come off as abrasive, but phone cameras are very good. You aren’t going to beat them optically for $200. The biggest shortcoming is stabilization vs mini cameras on a gimbal, but you can get around that with Gyoflow for free:
https://docs.gyroflow.xyz/app/getting-started/supported-cameras/mobile-phones
Check out the Blackmagic video app, too.
We have older phones, and the video quality isn't that great, especially in darker environments. I'll try Blackmagic, thanks for the suggestion.
…Also, I just realized you have a third option.
If you’re pretty technical, there’s this Canon firmware hacking software called magic lantern:
That you can load onto a classic EOS M:
https://en.wikipedia.org/wiki/Canon_EOS_M
The M and its now obsolete lenses it are cheap, and magic lantern gives it rather excellent video recording capabilities. Smaller cameras just can’t compete with its big sensor.
But if you don’t consider yourself much of a tinkerer, I’d avoid that route.