Nah, if it winds well I wouldn't look for the belt in the first place. It's more likely the tape path, either the pulley wheel or the capstan shaft or engine (like the capstan bearings, they might need some sawing machine oil). A loose belt would not sound winy, it would just sound pitched down. But it can be both, I can only guess from here.
This is what sucks about 90% of all those older recorders you find, either reel recorders or cassette recorders: they are often in a horrible condition, even the ones that look all pro and cool and stuff. They should be kept clean, maintained and running. Most people never really did this, they would just consider the recorder broken and replaced it with a new one. My dad was one of them... But I can't blame them, one certain problem can have different sorts of causes, it often wasn't worth the technicians research fee...
Oh, and I have to admit I'm like that too I have an old Philips PMC100 with a busted tape player, but I won't even think about repairing it. Maybe when I'm utterly bored, but definitely not because I think it's good fun