Inspired by Brian’s question here, which seems to have been answered with a pretty solid No, my follow-up question is whether or not allowing this to happen even though the rules do not would cause any significant problems at the table. Does this open up any serious abuses? Or, on the flip side, will the Rogue be unable to keep up?
Personally, I don't see the harm in it, and in fact I think you're probably losing more feat support by dropping the crossbow than you are by picking up the short bow. There is a lot of excellent feat support for a rogue cross bow user. The only real benefit I see is access to the "Halfling Short Bow Hunter" feat as well as maybe "Sly Hunter" at paragon.
The point about magic arrows vs magic bolts that Joshua makes is interesting, but they are rarely enough used and mechanically underpowered (don't stack with magic weapons) that I think that's safe to ignore.
To address whether or not this is a substantially weaker trade off, I don't think so. Again, weapons matter little to a rogue, their damage typically relies on the ability to apply that sneak attack damage. Typically short bow would be weaker than hand cross bow by a pretty significant degree, but making this houserule narrows the gap to a degree (repairing the to-hit disparity). The disadvantages I see mostly show up in paragon tier (you lose the ability to dual wield hand crossbows, you lose the extra attack on crit (both from two-fisted shooter).), and neither of them is particularly huge (the second attack only happens 5% of the time at paragon and 10% at epic). I don't find the trade to be a substantial one, though it is slightly weaker mechanically.
The tradeoffs seem like this is a reasonable exception to grant. To me the hand crossbow is a far superior weapon for a rogue, and this doesn't close the gap enough that I wouldn't make an exception for it.
Maybe, the weapons have identical statistics, but I think the biggest abuse is that the type of magical arrows available are different/more than those available as bolts.
There are exactly 10 different kinds of magical bolts available whereas there are 28 different kinds of arrows. Bolts tend to focus on adding control effects to attacks against enemies or giving the PC some kind of movement or bonus. Arrows seem pretty focused on adding different damage types, more damage, and ongoing damage.
I think it could be a source of all kinds of social problems, unless it is a rule since before character creation and is not requested by anyone.
Sentences like "Why did you gave avdantage to the archer? Why don't you give some advantage to me too?" become possible in such a situation (and can be avoided only with a bunch of mature players, which isn't always the case). And yeah I see it as a sort of (DM) abuse, even if that's not what you meant with your sentence.
That of course holds true of most rules that are not estabilished since the beginning. The easy way to deal with that is to ask if everybody is ok with the change and specifically say you're not gonna make other changes like this unless everyone at the table (including you) thinks it's ok to do.
As for the balancing issues, I see none.