You've asked these question already several years ago and back then, you didn't seem receptive to the advice you received. However years later you ask for advice again on that same issue but you already concluded that the advice that folks have suggested to you does not apply. Interestingly, you personally can not provide specific details because only your "friend" whom you trust worked on it. So if you trust your friend who tells you that the belts are good, and I'm presuming that you've changed all 3 (if memory serves me correct), and I'm presuming that all the pulley and belt surfaces were properly degreased and cleaned, and that everything was properly lubricated (including the solenoid to ensure free movement) as well as head transport assembly, and your friend tells you that it's a gear or cog, and I can already tell you that the cog or gear is already no longer available, then what advice do you want us to give you?
Some of the symptoms you are experiencing are common with slipping belts with this model. But you conclude that the belts "can't possibly" be the issue. I've seen belts slip on this model with new belts if they are just a tad too small, or if the belt transport surfaces are glazed or slick (easy to do when handling the belts since they often come in contact with greased parts during the install). Also, during some parts of the mechanical servo action, there are significant torque humps that the motor and belts must overcome. This includes start and stop, and head actuation action. The belts assembly could work fine until it comes up to the torque hump which it might fail to overcome, in which case it may struggle.
Also, you've discounted any possibility of fusible resistor issues. HOW DO YOU KNOW? You said you aren't technical and even afraid to just remove the screws. And yet somehow you can conclude that this common issue likewise does not apply? Let me just tell you that the fusible resistors in this (and many other boomboxes) have very low values and run hot. Over time, the values change, almost certainly higher WHICH INTRODUCES RESISTANCE INTO THE CASSETTE POWER CIRCUITS. There are like 4 of them. Sometimes they fail open and in that case, certain functions won't work. However, they don't always fail open, and if the resistance is too high, then the additional resistance might be just enough to prevent the functions from working reliably. And by the way, the solenoid and even the capstan motor are protected by fusible resistors. Do you think those 2 items I just mentioned need full power in order to function properly? After all, isn't the capstan motor which spins the flywheel and provides the kinetic energy to overcome certain servo functions important?
Here's some observations. Firstly, the microcontroller does not allow more than one function to operate simultaneously. So if you press a button (say stop) and the solenoid is unable to engage, the controller continues to wait. In the meantime, no other function is allowed to be processed until the last function is successfully processed. This is why unplugging the machine is often able to reset the unit. Also upon power up, the machine always performs a reset. If it clatters during startup, it means that it is unable to complete a reset. Also the system also has a stop over-ride switch which actuates when you eject the door. This is to prevent damage to the heads if they are up when the deck is operating. If you don't believe me, just try FF or RW or Play and then press the eject button. The deck will instantly stop before ejecting the door.
Now, since YOU have the boombox, and you aren't qualified or able to work on the unit, and YOUR friend is the one who will do the work, then just go back to him. And if you can get him to work on it, then you should seriously consider asking him to to heed the following advice:
1. Check and replace belts if necessary. Just because they fit doesn't mean they don't slip.
2. Clean all belt transport surfaces and deglaze if necessary.
3. Clean all moving parts to ensure free movement, and grease or lubricate where appropriate.
4. Check that sliding levers are free and don't bind. This is very important because many functions depend upon free movement of these slide levers, even if their functions aren't immediately obvious.
5. Check and verify ALL fusible resistors with a low ohm capable meter.
6. IF you conclude that the issue is a bad cog or gear (which I don't think anyone here has ever seen), then either live with the condition or find a used replacement deck since you aren't going to find the gear from JVC.
Frankly, if you aren't capable of working on the unit, I don't understand why you keep asking for technical advice since you can't perform them anyhow. Perhaps it's better if your friend comes on here and provide details on what he did, and didn't do and how he came to certain conclusions. On this model, 95% of the issues can probably be sorted by just following those steps. And if you ask for advice again, be prepared to be asked how many ohms were those 4 fusible resistors -- because I certainly would be wanting to know.