Other than proceeding with your steadfastness to this desire, I have no recommendations.
I'd be more inclined to get a Tascam 488mkII and a couple low cost/batt op condenser mics and record in multitrack to tape. If I wanted more analog flavor I might step up to a 388, 34B or 38/M30 mixer. If I didn't care about multitracking more than 2 tracks, I might recommend a 124AV or Tascam 32. If I needed something to tweak the raw sound of this setup I'd consider a Fostex 3070 stereo compressor/limiter.
Oh wait,... I already have all that equipment.
Q: If the tape-play mechanism is broken, how did you determine that this boombox provides the "sound" you're looking for?
To answer you basic question,... in the turnaround,... have you considered just cutting the wires to the motor or finding a connector point to disconnect? I don't think there's any clear indication up front that the whooshing sound is coming from the motors at all, rather than from the amps. You will not know this for sure until you go in deep cutting and slashing, and if this boombox is nothing to you other than an expendable project box, then I'd do just that. You don't know what you'd get til you try.