"never did" vs "never does"

Solution 1:

"Never does" and "Always does" in this context indicate that these are ongoing habits or traits that you have seen, continue to see, and presumably expect to continue into the future.

The past tense of "Never did"/"Always did" says that they were happening in the past, but does not necessarily mean that you expect them to continue. It may be that the person is deceased, or that you no longer know them, or that you are just not saying.

Solution 2:

The key here is the future tense. "He won't speak up. Never does." means: we can be sure that he will not speak up because that is his custom. The present tense here covers all the timeline: he has never spoken up and he will maintain this behaviour.

If we use the past tense, on the other hand, we lose this continuity and the sentence sounds awkward.