How can frequency dispersion in a vibrating string be modelled using an LTI filter?
From your question, it sounds like you think the all-pass filter is an independent post-processing step that takes as its input the output of the classic Karplus-Strong model (delay line, LP filter, close the loop). If that would be the case, yes, it would be very surprising indeed that new harmonics would appear! But here, the allpass filter is part of the closed loop. You can intuitively think about what it does in these terms: it is a short delay, with a delay value which is frequency-dependent (phase shifter). This gets added to the delay provided by the main $N$ samples delay line, and causes the lower frequencies to "see" a longer loop. So the highest harmonics will have a period close to $N$, and the lower harmonics will have a lower period. The overall transfer function of the loop is no longer a comb with regular spacing between teeth, but a comb with a slowly increasing gap between teeth.

What you are forgetting also is that this whole system (delay line, all-pass filter combined with loss LP filter, close the loop) which is a LTI, is excited by a burst of white noise. So the input signal contains all frequencies, and there's nothing surprising or contradictory with the fact that altering the transfer function of a LTI can change its frequency response so that its resonant frequencies are shifted, or so that new bumps appear.

If you need to convince yourself... System A is a band-pass filter with a frequency of 1kHz. System B is a band-pass filter with a frequency of 1kHz ; in parallel with a band-pass filter with a frequency of 2kHz. Both are excited by white noise. In the second case you have "added" a LTI filter in the system; and a new frequency has appeared. There's no magic in this - you have just shaped the frequency response of the system differently to sculpt the input in new ways.