My record player sounds slightly high pitched

This thing is pretty handy -  https://www.turntableneedles.com/Digital-Speed-Readout-Turntable-Platter-Speed-Tool--Any-Table_p_3987.html

Probably not perfectly accurate, but it's been a big help with some of the tables I've had with a detached motor.

I agree with the advice of a new TT.

The iPhone app is known to be inaccurate. In a situation where you really want to know what is happening, it is not the best tool. Anyway, the OP seems to have lost interest in his own problem. So we need not get excited about the solution.

Just see if you are close on the speed. Download the turntabulator app on your phone, it's free, and see what it says. Not as precise, but close enough for diagnosing your problem.

Tip, put your phone on a roll of masking or other tape so it can lay flat over the spindle.

Hi,
higher pitch = higher speed.
Try a strobe disk to see how far out.
Belts with age tend to slower speeds so could be a drive issue.
Does your tt have pitch control or is the speed knob working properly?
I would care for a better tt.

Start saving for a better turntable. Aim for a Pioneer PLX1000 with Nagaoka MP110 and Schiit Mani phono amp. 
Below that, a Project Debut Carbon/Audio Technica AT95 would be a decent starter table. All those cheaper ones are junk. 

High pitch = Fast

Do you have other sources to compare to?

Download and print this free strobe disk and use under fluorescent light to determine if this really is a speed related problem.
https://www.soundsclassic.com/ttsdteg.html

If the belt is or was slipping, I would have guessed that the music might sound "flat" or slightly lower in frequency in other words.  Not "high pitched".  I suspect that the turntable may be running too fast, at a speed greater than 33.33 rpm. To test this, the cheapest accurate method is to purchase a strobe device.  The very best strobe device is that made by KAB Audio; it's a complete kit, but it is not cheap. On the other hand it is a very worthwhile piece of test gear to have around.  You can order one from the KAB website.