I want to challenge your approach to this. As you note, the Light property is defined like this:

Light. A light weapon is small and easy to handle, making it ideal for use when fighting with two weapons.

This property isn't just there as a computer function — it's also part of the basic description. A hand crossbow is small and easy to handle. And it is, as far as crossbows go, ideal for fighting with two weapons. You may need special practice and training (as represented by the feat) to take advantage of it, but it's undeniably better than any other crossbow for that case.

Is it true that mechanically there are no current rules which make particular use of this property? Meh. Probably so. But it does not mean the property shouldn't be there, because that kind of mechanical interaction isn't the only reason weapons have descriptions.

It might have been more elegant from a "the puzzle pieces must fit together!" perspective if Crossbow Expert said "a crossbow with the light property" instead of "hand crossbow", but I don't think that really would add much of actual value.

Of course, there's always the possibility of future features which apply to "light ranged weapons" or the like.

The trait states that you can only fire one piece of ammunition, not that you can only make one attack. For example, someone could use one attack to fire the weapon, and a second one to use it as an improvised weapon to bludgeon an enemy.

Given that you're already homebrewing this weapon with special properties, it's not a stretch to homebrew your own trait for it. Why not add something like:

Unwieldy. Because of the awkwardness of this weapon, you can only make one attack with this weapon per round, regardless of the number of attacks you can normally make.