Well, all I can say is you have ruled incorrectly. The game designers have weighed in, and they don't agree with you. Either way, it's really not a big deal.
We'll talk about that at the start of our next gaming session together. I might not be able to accommodate your wishes completely, but I am sure we can come up with a compromise that works for both your character and the campaign world. If Mike Mearls shows up with his janky Gnome Illusionist/Barbarian, I'll be willing to hear his point of view as well.
And I think that is a rather disingenuous line of reasoning. You honestly think that they allowed the duration of 24 hours on the off chance that that the PC's may have to go a full 24 hours with no rest? And just those two spells? Please.
No.
You asked why else the duration would be 24 hours, and I gave you a reason why the duration would be relevant. While you might have difficulty imagining a group not being willing/able to take short rests during an adventuring day, that speaks more to the strength of your imagination than it does to any cogent rebuttal.
You might say that having a short rest in a day is pretty common, and use that as justification for why concentration spells with a 24 hour duration ignores short rests. The flaw in that reasoning is that a 24 hour period is significantly more likely to have a long rest than it is to have a short rest, and your logic would apply equally to those spells ignoring long rests.
After all, since a long rest is even more of a given than a pair of short rests, resulting in an adventuring day that is only really 16 hours long, your logic would apply even more strongly to concentration not being interrupted by long rests.
As is, since a short rest doesn't incapacitate the caster of a concentration spell, there is no reason for a caster to be unable to concentrate on the spell during a short rest.
You cannot benefit from a short rest if you do anything more strenuous "eating, drinking, reading, and tending to wounds." Whether or not short rests incapacitate the character is not the issue; the issue is whether concentrating on a spell is more strenuous than "eating, drinking, reading, and tending to wounds."
Fighting, engaging in exercise, and participating in a juggling routine do not incapacitate you, but they all are more strenuous than "eating, drinking, reading, and tending to wounds."