Definitely don't buy an RX 5500XT. A GTX 1650 Super for $50ish less is a far better purchase, or just spend the extra $20/30 over an 8GB 5500XT for a GTX 1660 Super.
An RX 5500XT may be slightly faster than an RX 580 and more efficient, but it's also now twice the price that you should be able to fetch a 580 for.
Don't just buy "any 80+ Bronze PSU" either. Plenty of them are absolute trash. Get a good quality PSU. That said, PSU pricing is horrendous right now and the cheapest PSU I can find that's actually reasonable quality, is $80.
For motherboard, if you want a budget AM4 board, get an AsRock B450(M) Pro4(-F). If you're spending more than that, look at an MSI B450 Gaming Plus(MAX) or Tomahawk(MAX). Availability for motherboard is pretty awful right now though, pricing is not as bad as PSU's but pricing also a bit iffy.
$100ish for AM4 CPU does indeed mean R3 3100/3300X or R5 1600AF/2600 are your options. I'd personally go with a 1600AF/2600 over a 3100/3300X but that's up to you(and stock/pricing).
16GB RAM is a no brainer, 3000MHz minimum. You should really be looking at a 3200MHz kit IMO. If you can manage an extra $10 or so for a decent 3600MHz kit though, that would be ideal.
Not all RAM is equal, just look for whatever has decent timings for the frequency you choose. e.g 3600 @ CL18 or 3200 @ CL16. Patriot Viper stuff tends to have decent RAM(not sure if it's binned or not), but the Viper and Viper Blackout stuff is usually pretty good for manual tuning. If you're just going to set XMP(A-XMP/DOCP) and be on your way, then yeah, it doesn't matter so much. Just get [email protected] CL16 or 3600 @ CL18 and you're good to go.
Phanteks P300A, P400A, and Cooler Master NR600 are the budget cases I'd be considering. If you go cheaper, you start losing features/quality.
Don't cheap out on SSD. Don't buy the cheapest SSD you can find. I'd be aiming for a half decent 480-512GB SSD, like a Crucial BX500(or similar). If you can manage to up it to a 1TB SSD, just grab yourself a nice 1TB NVMe SSD, like a Crucial P1/P2, or even better, a Sabrent Rocket(Q) if you can get them for a decent price.