Post me a pic of the entire pickup deck mechanism; If I can identify it, I may be able to give some more specific pointers than the generic ones below...
The rubber bobbins support the entire CD deck mechanism; altering them won't alter the relationship between laser head and disc, which has to be pretty much dead-on within a few thousandths of an inch for the focusing element to be able to keep up with inevitable disc warpage and runout.
Manufacturing precision of optical media is not that great; the optical pickup has to be able to move up/down +/- 1mm or so in sync with warpage of the disc to be able to function. This means that being off 0.20-0.030" can put the surface of the disc outside that range, and only a few very flat discs will play. I used to keep a stack of discs I'd found and labeled that were "Extra flat" "Average" and "Extra Wavy" for testing.
Another thing you need to watch out for is the magnetic platter clamping disc being mucked up; often they will have a rare-earth magnet doughnut inside, and that rots with age and deforms the flat surface so it doesn't lie flat, or in some cases actual loose flakes of the magnet can cause mayhem the mechanism.
Compressed rubber bobbins are a bit of a PITA to fix; I've had more than a few units that I had to chunk because the bobbins weren't available to buy ANYWHERE and I was unable to scavenge suitable replacements due to some oddball design element. Some other options are: Old PC optical drives, o-ring assortment as spacers (rarely work; usually too thick) and making washers out of craft foam sheet from Wally World (the stuff they sell nowadays instead of craft felt).
These things (and the magnetic platter clamp) are to CD player mechanisms as capacitors are to electronics in general; "the 3¢ part with a finite lifespan that brings everything down eventually". They know this, and as it is a custom-molded part, it is easy to limit production, thereby ensuring turnover. They don't want your favorite CD player becoming an heirloom piece like your favorite Marantz 2xxx or 4xxx series receiver did.
What is key is to make sure that when the disc and platter are engaged under the magnetic clamp, that there is adequate clearance above the loading tray that it doesn't hit the tray, and that it appears level, and that if you depress the spindle gently, it can move down a couple mm before the disc bottoms out on any edge. If you can get it to that point, you should be golden, and the unit should be fairly vibration tolerant once reassembled. It's usually a lot of trial/error fiddly-bit work though, and often the special shoulder screws go into plastic, which strips out easily and deforms a little with every screw/unscrew cycle, so confounding your efforts.
And sometimes the laser head just gets weak... the diode laser dims or shifts frequency enough that the photodiode doesn't read reliably, or the drive current shifts enough due to aging of electronics that that the laser is over or under driven... or maybe this week solar flares are kicking the ass of everything electronic, and you just drew the short straw on the great circular workbench in the sky.
Point being that if you can't get it to play reliably, even after hours of your tenderest ministrations, don't be too disappointed. Optical drives, especially those made for CD audio, are made absolutely as cheap as possible with the intent of only outliving the warranty. If you get 10 years out of one you're lucky; by which time the deck is often no longer available as spares.
That said... once you DO get a CD player sorted so it works correctly again it can be most satisfying... until the next time it fucks up.
Good luck!
mnem
You're gonna need it.