My girlfriend's ex recently got a decent PC with a 5500XT inside for something like $600 at WalMart. He too was trying to game on an aging PC (from around 2007) and asked for some guidance.

These days. most PCs come with an adequate CPU to game on. I would aim for a 6 core if you can swing it. The internal GPUs will be better than that laptop experience, but not too much (in terms of Valheim). I havbe a fairly new 6 core laptop with a decent iGPU and it barely plays Valheim at 20 FPS. If you can find a deal on a PC with pretty much any dedicated GPU, you should be in good shape. This is assuming it's going to be played at something like 1920x1080 or 2560x1080. 1440p and up needs more GPU.

If I google "5500XT gaming PC", I get links that show some 6 core CPU PCs paired with RTX3060 and 5500XT for $650/ $700. Many in that price range only have 8gb of ram so I would upgrade that to at least 16gb (or buy 16gb) or look for systems with at least that much to begin with.

I would say anything with a relatively recent 6 core CPU, 16gb of ram and some kind of video card can be had for your budget and might even be good to go for some time.