Sometimes you can clean the spindles with ISO91+ to get them to grip again however, I noticed the crack/split in the rubber on the supply reel so you are correct in needing replacements. Play is done with that tiny spindle on the bottom of the take up reel. FF is done with the center black plastic wheel rocking to the left and spinning the take up wheel. Rewind is done with the black plastic wheel in conjunction with the idler above (to reverse rotation) to the supply wheel for rewind.
Because play is fine but FF doesn't work, I am beginning to suspect you have more issues than the rubber on the reels. We can visually see that the rubber is split on the supply reel so that's a given however, that black center wheel is driven by a plastic clutch wheel which has its own rubber idler and uses springs and mechanical movement to provide friction against the flywheel and a slipper clutch to transfer torque to the reels. If its slipping you will loose FF and Rewind as well. The torque is adjustable on both playback and Rew/FF clutches by rotating the metal tri-collar cog wheel but these are set at the factory and should only be adjusted if you have means of reading the take-up and supply torque specs.
In addition to the idler tires getting hard and brittle, is the clutches plastic retainer cap which retains the spring tension for the clutch pressure plate friction. Those nylon plastic collars are known to split causing the clutch to slip. If this happens on the FF/Rew clutch you will see/hear it rubbing on the underside of the flywheel.