A few minutes ago I clicked "buy it now" on a refurb Panasonic Lumix DMC-FZ300 on Ebay for $297 plus tax, putting it just above your price range. Yes, it will shoot 4K video, but I picked it up mostly for it's still camera features.
Yes a phone will work as long as it has a decent camera, but when I started out on YouTube, the simple auto-everything nature of a basic camcorder helped me concentrate on content, not production, and looked better than my old HTC phone's camera.
I started with a Panasonic HC-V180 that I found for $150. It shoots 1080p, has "intelligent auto" and good stabilization, but I was not always happy with the built in microphone, so I stepped up to a used Panasonic HC-V770 that I got for $275. It has an external microphone input, better manual controls, wifi connectivity, and other features, but again, 1080p.
I eventually went 4K with my Panasonic HC-X1000, but this is beyond your budget.
I know a lot of folks think camcorders are outdated, old school tech, but they make shooting video super easy, and this is important when you are first starting out.