Firstly, if you search for ...

sony tc458 reel to reel tape recorder

... a number of relevants eBay items come up - belt kits, etc.

Secondly, I am now beginning to remember that my machine had exactly the same problem. I can't remember the model number of mine, but it was a dual R2R and AC machine - two decks on one chassis with the stereo speakers clipping together to form the lid for transportation. I bought it in 1973, and this failure occurred within a couple of years, so nearly 40 years ago, and as I fixed it successfully and only disposed of the machine around 2000, and myself am in my 60s with all that implies for memory, I believe I can be forgiven for not remembering all this sooner.

Of course, your model may be different, but this is how I remember the details of mine ...

The take-up spool was driven not by a belt but by a rubber 'idler' pulley (I suspect the belt that you've noticed merely drives the tape counter). When the tape was at rest, the idler was pushed clear of the drive and driven pulleys by a rod connecting with a cam on the control knob spindle. When the control knob was turned to 'Play' the cam, and therefore the rod, receded, allowing the idler to engage with the pulleys under the action of a spring. As I recall, the idler pulley was mounted on a sliding plate between which and the chassis was some grease that became too stiff for the manliness of the spring, leading to the exact problem you describe. I have a vague recollection that I took out the pulley, plate and all (by removing some nuts, circlips, etc), cleaned the grease off its underside and off the chassis, and replaced it with vaseline mixed with a drop or two of Singer oil. I may also have replaced the spring with something beefier.

AFAICR, I never had the problem again.