Hi Stefan .... welcome to VE!
A SL-23 is definitely more worth it than a $100 USB turntable. I don't think anyone has pointed it out yet (odd!), but your user and service manuals are here: https://www.vinylengine.com/library/tec ... l-23.shtml
Couple things to note:
1. Speed Adjustment on page 3 of the service manual refers to the variable resistors (VR1 & VR2) located on the speed control board inside the turntable (not very clearly identified, but you can see them in the circuit diagram on pg 6). I'm not sure if there are holes in the base or bottom board to allow you access to these, but otherwise remove the bottom board and you should be able to locate them.
A warning ... remember when you adjust these you have the turntable powered-up and running. Be very careful!.
2. Clean the variable resistors, 33/45 speed switch and the pitch control pots (the ones with the external knobs) with contact cleaner as has been suggested. You have to get inside the pots and switches (look for small holes in the sides you can get the spray-straw into). The resistors are typically exposed so just spray from the top. Exercise the switches, pots and resistors 50+ times and then repeat spray and exercise again. Let dry. Many contact cleaners have lubricants added. Some (usually identified as 'zero residue' do not). For example, if I use Caig Deoxit to clean then I follow-up with Caig Faderlube to lubricate.
3. Then, with the new belt installed, set the external pitch control knobs to be close to the mid-point of their adjustment range and then adjust the variable resisters to set the speed correctly (using the strobe marks on the platter). This will give you the maximum +/- adjustment range once you have everything buttoned back up.
4. Sewing machine oil is just fine to use for lubricating the motor and center spindle. However, since you probably have the turntable bottom off I would pull the center spindle completely out and clean (isopropyl alcohol) and re-oil the spindle and the bearing well. After all this time the oil in the bottom is going to be black goo. There is a small set-screw in the side of the bearing well. Remove it and the spindle should pull out from above. Careful - there is a ball bearing in there as well. I've had to replace them on some turntables that have sat a long time ... look for replacements at a bike shop.
5. Also since you have things open, look at the auto-return mechanism (you have confirmed it works ... right?). It might need old dried lube cleaned off (alcohol) and new grease applied. Use a plastic-compatible grease (is there a hobby store near by? ... they often have good greases used for model trains, etc.). I use Labelle 106 ... but not sure if it is available to you or not. Only apply new grease where you see old grease was used.
6. Check the cueing. Does the tonearm drop slowly to the record and lift properly at the end? You can test this out as you are testing the auto-return. Install the cartridge, but not the stylus, and put the counterweight on so you are getting some downforce. Don't put a record on the platter. If the tonearm drops like a rock that's not going to be good for your new stylus (or records!). You'd need to replenish the silicone oil in the cueing mechanism. That's a bit more involved ... so check it and let us know if you have to do it.
Hope that helps!
By the way ... the use of 'vinyls' to many of us is like running your fingernails across a chalkboard. LPs, records, vinyl ... but not vinyls!.