I wouldn’t bother scrubbing the bed up, it will come up fine after a couple of weeks of use.
Keep it clean and oiled and it will soon sort itself out.
I’ve never needed to remove my spindle.
Take the guards and the backgear off first, strip all the easy stuff off the unit.
These kind of bits fit back on really easily and would appreciate a clean and an oil.
The headstock is pretty complicated compared with the rest of the lathe, it’s a solid casting with some pretty serious precision parts.
There is the bullgear at the front and the engagement pin, which always drives and is always engaged/keyed to the spindle, while the pulleys are locked in/out of engagement via the bullwheel pin.
You can undo grubscrews on the pulley but the bullwheel is a bit of a mystery, my guess is it has a keyway and the spindle gets tapped out from the back.
Two sources I can think of.
These guys will DEFINITELY know what to do.
Sign up and ask them.
And of course there’s always tony at lathes.co.uk