PS1 turns on fine after my first ever soldering job but discs won't spin - what should I check?

The silence you’re hearing is usually a sign of a connection failure somewhere in the power chain, not just a bad joint. Since the system is booting and the SD menu appears, the main board is likely happy, but the new components on the card aren't completing the circuit needed to spin the drive motor. This almost always points to either a bridge that is too subtle to see with a magnifier, or a cold joint on a critical power path. If the connection is intermittent, the drive controller sees fluctuating power and will just refuse to initiate the motor to prevent damage.

Try to isolate the power supply lines going to the drive cradle. Even if the visual inspection is clean, you might have a poor connection on the ground pin or the +5V line. Sometimes when soldering complex PCBs, you might think a joint is good because it looks shiny, but it has microscopic voids inside the solder paste that prevent a solid connection. You need a reflow or a simple reheat of the connections to ensure the metal is properly wetted.

If you've checked the power lines and they look solid, then it is very posible you have inadvertently created a very slight short. These are hard to spot with just visual inspection. What you need to do is apply a very light heat to the individual pads and look for a tiny, almost invisible blob of excess solder bridging two nearby points. A de-cap test, where you carefully peel back a bit of flux and visually inspect the pins, can sometimes reveal these issues. If all the solder joints look perfect and the power connections are confirmed solid, then the drive likely wasn't the problem, or the damage was too extensive for the mod to save it. Sometimes these older components are just past thier expiration date, and the X-Station mod was the final straw that caused the drive to finally give up the ghost.

I'd push back on the 'microscopic voids inside the solder paste' bit. If you hand-soldered with wire solder like the OP describes, there's no paste involved at all, so that failure mode doesn't really apply here. Shiny joints from wire solder are usually fine. What I'd actualy check first is way more boring: the lid switch. These consoles won't spin the drive if the console doesn't think the lid is closed, and it's easy to knock that switch out of whack while you're in there. OP said the drive was already flaky before the install too, so honestly a dead spindle motor wouldn't shock me either. Does the drive do anything at all when the lid switch is pressed down manually?

When I did my first mod on an old handheld, the soldering was actually fine but I had knocked one of the thin ribbon cables loose while working. It looked plugged in but it had slid half out of its little socket. Pushed it back in and everything came alive. Maybe check that the drive cable is seated all the way down where it meets the main board? Also might be worth trying a disc you know is in good shape, since a scratched one can act kind of dead on a drive that was already weak.

The lid switch thing is real. I did a mod chip install years back and spent two days convinced I'd wrecked the board, turns out the little plastic switch actuator hadn't clicked back into place when I reassembled the shell. Console treated it like the lid was open the whole time and just refused to spin anything. It looks fine from the outside so you'd never guess. Take the top shell off again and press the switch by hand while it's powered on, see if the drive wakes up.