I'd go for a 10 minute short rest rather than 5 minutes, because of the way the time interacts with other mechanics. In particular:
1) What you can do during a short rest. EG: Casting a ritual spell (takes 10 minutes) or certain other spells such as Prayer of Healing (10 minute cast time), which is reasonable to want to do during a short rest.  A 1 hour short rest allows casting something like Resurrection, though I feel that is out of scope of a short rest action.
2) Whether you can use effects that were initiated before the short rest, after the short rest. A 1 hour short rest means any spell effect of 1 hour or less is guaranteed to have expired (for example: Pass Without Trace, Invisibility). A 10 minute short rest would allow those effects to still be active, but 10 minute duration effects would have expired (for example: Fly). A 5 minute short rest would allow 10 minute effects to still potentially be useful after the short rest.
My feeling is that 1 hour duration effects should still be useful after a short rest. I don't particularly feel that 10 minute duration effects need to still be useful after a short rest, so the 5 minute break seems unnecessarily short.
Between the above factors, then, 10 minutes feels like a reasonable short rest time.  15 minutes would also give roughly the same effect. It's mostly just a matter of what sort of break times fit your mental idea of a "break". (5 minute break; 10 minute break; 15 minute break)  And apropos of nothing, it also gives a complementary meaning to "Take 10".
However, yes, making short rests actually short does increase the feasibility of just spamming short rests to recover abilities.  A 1 hour short rest makes it difficult to meaningfully get more than a couple per day if you want to get other things done. A 10 minute short rest is something you can take every hour without significant drawbacks, meaning you could have 10 short rests a day and barely notice.
Limiting the number of times you can use that to recover abilities may help keep things in check. Scaling with proficiency bonus, and getting the ability to restore your short rest abilities more times per day as you level up feels appropriate. If we go by Level Up's math, that's pretty much on par with the expected number of medium encounters a party could expect to handle per day: Tier 1 has +2 prof, and gets 2 medium encounters per day; Tier 2 has +3 to +4 prof, and gets 4 medium encounters per day; Tier 3 had +4 to +5 prof, and gets 6 medium encounters per day; and Tier 4 has +6 prof, and gets 8 medium encounters per day.
Of course this does imply that you should expect to get enough short rests that you can pretty much always have a short rest before a (medium or harder) fight, meaning short rest jobs should always be running at 100%. Is that desirable?  I'm not really sure, but I don't find an immediate reason to reject the idea.