I've got the same issue with a DRR-M30 (same mechanism, both service manuals available online).. Looking closely at mine it's the slightly-up-and-down-movable bits of the hubs that are 'sticking' down if they get nudged down by a tape during loading and not springing back up to properly engage with the teeth in the hubs of the cassette itself. Therefore while the pinch roller and capstan are merrily drawing tape through, the take-up reel isn't turning at all, so tape spools out onto the pinch roller and surroundings..
It seems the spring on the forward take up reel is weaker (different part number to the other spring), so it's that one that presents the problem... hence on mine reverse play and rewind is ok but forward play and ffwd are not.
I hope it just needs a stronger spring, but I don't know where to get one or why they would have originally specced a weak one. Also I'm suspicious of the hub mouldings themselves - I've taken them apart and cleaned and lubed to try and get rid of 'sticktion', and parts move smoothly in my hand, but as soon as i put them back in the machine there's a 'gritty' quality to the movement... so something's going on..
I haven't got to the bottom of it yet, but I've started a thread on tapeheads so I'm hoping with their help we'll solve this apparently common problem...