The filtering has to remove more than clock noise. One of the other things it has to remove is the artifacts creates by the sampling aspect. Even though it is "analog", what comes out of the BBD is essentially a continuous series of square waves, with no means of making the transition between one sample and the next smoother.
Hmmm, square waves....where have I heard of those before? Oh yeah, distortion circuits! The act of sampling, no matter how great the resolution is, creates a lot of audible artifacts when the sampling rate is low enough. Maybe one's latest USB audio interface samples at 96khz, but for producing a delay of just a few hundred milliseconds, the sampling rate of a 4096 stage BBD will often come down well below 10khz. So, not only is there a risk of hearing the clock generator, but there is an equally great risk of hearing all those harmonics produced when a 4khz tone gets sampled at 10khz...or worse.
The extensive lowpass filtering is intended to smooth that out in audible fashion. One of the things you will often see in delay circuits is the presence of lowpass filtering not only after
the BBD, but before the signal ever hits it.
The reason for this is that if you don't feed the BBD with the frequency content that risks generating such artifacts via the sampling process (what is referred to as aliasing), then you don't have to work so hard to clean them up after the signal comes out of the BBD.