I've done that with greasing the gearbox. Made no difference, really. Not done anything with bedding in the brushes. Is this done by disconnecting the motor from the gearbox and free-running it for a while? It's one of my remedies for dodgy Saab headlamp wiper motors. Running it in reverse too also helps.I recall posts about greasing gearboxes and bedding in the motor brushes in the past, might be worth a search, but that's getting silly.
One other mod that I'd almost forgotten about was to the reel spindle. I found the reel rather stiff to turn so I replaced the steel washer under the spring with a nylon one (which was also a tad thinner) and gave it a shot of silicone grease to lubricate it. Any other grease or oil would probably be OK. That made it a bit easier to turn. Maybe this is worth a try on yours?
Or make a spacer for the screw to lower the tension on the spring? Not tried that.
Can't believe the transformer would take a few seconds to fill with electricity.
I know what you mean, though. Neither can I.
However, I can imagine the addition of a big-ish capacitor might help if the motor is demanding more current than the transformer can provide during start-up. Then again, if my power-supply is rated at only 1.5A and it can get it going, then a 50VA (2A @ 24V) transformer should be able to cope. Or should it? I'll try to measure the current on mine over the weekend since I suspect the power supply can manage a bit of overcurrent for a short duration.