Replace speed trimmers and re-oil the motor bearings.
The center bearing will not be the reason for the speed problem unless it was once lubricated with WD40 or some other nasty stuff - but
it could have a drop of sinter oil to its bronze insert, if it looks dry.

Martin

Yes, I know that the motor barings can dry-out. The spindle bearing is dry. I have another Beogram, a 5500 that I will try too. Just like the 6500 because of its built-in RIAA (re-capped with many film capacitors). I have two re-built cartridges, both with sapphire cantilevers with a nude line diamonds. To my ears they seems to pickup a bit of surface noise, more than I can remember from many years of listening to vinyl.

Roger 

Dillen, what kind of oil should I use for the motor bearings? I have some oil for capstans of Studer/Revox tape recorders. It is a "Sinterlageröl". Can that be a good choice for the motor too?

Roger

Yes, that'll be fine. But don't add it to the outside of the bearing material, it needs to get
into the pores of the metal. Bath the bearings in the oil under vacuum.
Beolover shows how on his blogspot and he just made a video of it too.
He shows a DC version of Beogram 4002 but the principle is the same for your motor. 

Yes, I know the process Beolover uses.That is a process above my capabilities. Beolover charges $ 80 for the infusion. I plan to send my 4002 or 6000 to Beolover for a complete overhaul. 

I will pull out my Beogram 5500 and see if it have any speed problems. 

Fitted new drive belts (platter and arm) and now it is running without speed variations.

Still, both my 6500 and my 5500 with on outboard RIAA, give a rough sound. Both have restored cartridges, sapphire cantilever and fine line elliptical diamond.

Maybe the the tonearms need some attention? What should I look for? 

I have tried both turntables in various chains, three different RIAA stages for instance, two differnet replay devices. Tracking force is checked and I have tried different settings. I have also tried the turntables in different locations and support (light, heavy). There is a rough sound with higher modulations. Sent back the cartriges to Tonabnehmer Service in Germany for a checkup and I was told they are up to the specifications. I have not checked the vertical tracking angle but it looks like more than usual. These Beograms also have some circuit breakers and relays in contrast to more conventional players. Can they add distorsion? What about the tonearm sledge, is there anything that need attention there? What about the rails they run on?

Will try to upload an example. 40 seconds in this track, there seems to be tracking problems, https://www.dropbox.com/s/kctz41h5bliw3o3/Burglars%20Holiday.wav?dl=0