Attuning does not interrupt a short rest
Under the attunement rules, it says:
If the short rest is interrupted, the attunement attempt fails.
If the process of attuning to the item itself is enough to interrupt a short rest (ie, because attuning is not "light activity" as required by resting), then the item remains unattuned after the short rest as well, as stated in the quoted rule.
This means nobody can attune to anything because attuning interrupts itself: it is non-light activity you must take over a period of time that requires light activity (ie, it's a contradiction, this rule is nonsensical).
So we must conclude that attuning to an item does not interrupt the short rest, and therefore attuning to an item must fall under light activity, as otherwise the design of this mechanic prevents anybody from using it.
If the short rest is not interrupted, they gain the benefits of a short rest
If attuning to an item does not interrupt a short rest, and nothing else interrupts it as well, then a PC can gain the benefits of such a rest. This includes recharging abilities, using features that key off short rests (eg, Inspiring Leader feat, Bard's Song of Rest), and, yes, using Hit Die to heal HP.
There are no disadvantages listed while attuning to an item
Focusing on an item during attunement does not impose disadvantage to Perception checks while they are on the lookout during a long rest. There is no mention of this in the DMG. But, the DM is free to impose any disadvantages he feels necessary as part of the focus attuning requires.
Impact to the game
I feel that if my DM imposed this penalty on me, I would just ask the members of my party to attune to items over short rests only, and to never do it during long rests. And then this penalty would never come into play, which makes it, on the whole, an immaterial penalty (unless an adventure is specifically designed with this in mind).