I am not familiar enough with open reel tape use to speak to degradation or potential noise/distortion from playback machine issues, but to me this sounds like a digital problem for the following reasons:
- It only happens at or around waveform peaks - why nothing in quiter or even 'silent' sections? I'd expect tape degradation to not correlate with signal so much (but as I said, not so familiar with that). This could also be distortion on the input chain of the recording process of course, but you say this was not there (from memory?) when first mixed. I suppose it could be tape player electronics on playback when they were digitized, but in that case it is still the fault of the dude who did the job and a bit of a hair split. The vibration over the tape head thing would also be more constant/random vs the signal wouldn't it?
- the nature of the sound is somewhat digital as well. Not straight up converter clipping, which is very much more obvious (and inexcusable), but sounds a lot like when a DAT glitches out or my old sound card when it lost synch on mixdown (hasn't happened since I was using a PII with the very first commercially available USB interface).
- the glitches are so quiet (like Pilgrim, I had trouble hearing them until I switched to better speakers) that it sounds like someone else may have already made attempts at removing them. This is not a strong point, but a possibility worth mentioning.
- the audio seems to be rather compressed, even when just looking at the waveforms on Soundcloud, but it sounds like it too. A clean vocal track will never be that narrow dynamicly, especially when belting it out like that first bit. It could have been on the input chain (compressing the shit out of everything was very much in style in LA at the time), but if it was done after, especially if in the box with plugs and if one of those (multiple daisy chained compressor/limiters are all the rage now) was a brick wall limiter, then there's a potential culprit too. Plugins or other DSP can leave artifacts that would not be as loud or jagged as proper clipping, analog or digital.
The only way to know for sure is to go play back the tape on a known good properly aligned deck and listen for it. Was the deck used in the digitization process the same deck it was recorded on (maybe even yours)? Did you perform the task (i.e. know exactly what was done and what was used to do it)?
As for what to do about it, hard to say. The problem is the DAT-like slide nature of the noise. I'd start with a parametric EQ - something surgical and capable of extreme notch filtering. You may need to cut multiple freqs or use a slightly wider Q or band to get the noise out, but it might not cut it. I'd then try any sort of denoiser/pop remover you can get your hands on (e.g. for cleaning up digitized vinyl), but it's hit or miss with these and different ones work better on different types of noise. I have heard good things about Spectral Layers but have not tried it myself.