Let me address my issues with the equipment, as I don't want to seem dismissive without consideration.
1. Sony CD Player (CDP-C321)
This is probably fine; but it's ancient and because it lacks a digital out, you are relying on its DAC. Likely you can continue with this happily, but the cost of a used replacement that does have a digital out is so trivial it's hard to argue against replacement.
2. Turn Table Technics SL-BD20
I have no real comment on a turntable. I suspect any replacement would blow your budget
3. Sony FM Stereo/FM-AM Tuner –ST-JX521 (and attached Sony Integrated Stereo AV Amplifier (TA-AV621))
The AVR is low-end, ancient and of a brand not known for quality amps. It's also from the pro-logic era.
A decade newer, higher end, Digital 5.1 used AVR is very inexpensive. Given the use (audio CDs) it's also more than sufficient.
4, 2ea - Front Floor Standing Speakers –Sony SS-U621AV (8 ohms/270 watt)
Sony was not known for the sound quality of its speakers. Between driver design/layout and cabinet resonances; I suspect (I have no personal experience with these speakers) you would be well served by a more modern enclosure. Given your price range, that's gonna be a 2.1 setup.
5. 2ea - Rear Bookshelf size Speakers- Sony SS-U31 (8 ohms/30 watt)
6. 1ea - Center Bookshelf size-Sony ss-CN62 (4 ohms/60 watt)
Not useful for CDs or records; and (I'm assuming) from the pro-logic era so simply not worthwile as center/surrounds.