clearstream
(He, Him)
They could complete a short rest, and recover some HP. Just not all. However, for me the more helpful point comes from thinking about rests as a foundational mechanic. Through moderating resource use, rests open up design space (i.e. for abilities at different power levels). Thus, it shouldn't be too hard for a DM to disrupt a rest. If effectively any amount of combat won't do it, then it becomes a lot more difficult for a DM to say exactly how a rest would ever be interrupted.It is also ridiculous to assume that casting a spell (something PCs do all the time, considering cantrips are at will) or fighting for 1 minute (10 rounds) out of 8 hours of resting won't allow you to gain the benefits of a long rest.
Imagine this IRL, a bunch of soldiers are in a fight, get some downtime to rest, have a brief skirmish, and then continue resting to the full time. So, these soldiers now recover no HP?
In the past, I have been persuaded that the two readings are both permissible: the text is literally ambiguous. One might want to say that short rests and long rests are categorically different, so that things that impact one may or may not have any impact on the other. That is not my position, because they seem in so many respects similar and they satisfy a connected mechanical purpose. Thus, I believe that if casting one spell can interrupt a short rest, then that implies that the casting of spells that can interrupt a long rest must be as few as one spell. That of course then implies that the "1 hour" is tied to "of walking" and does not span the sentence (which FWIW strikes me as less ridiculous when you consider the full scope of cases arising).