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.

You load as part of any attack with a crossbow. You can draw the ammunition as part of the attack, so loading it is implied to be part of the attack as well. "Loaded" in this case means only that you have ammunition available for the attack. Per the official errata, "loading a one-handed weapon requires a free hand."

This means that, post-errata, using rules-as-written (RAW), you cannot indefinitely fire two hand crossbows in the same turn. If you have a single attack, you can fire once with your action and once with the second crossbow as your bonus action with Crossbow Expert on the first turn of combat if they start loaded. However, you can't keep dual-wielding because you won't be able to reload with both hands full.

If you stow the second crossbow on the first turn with your free object interaction, you can fire the main crossbow twice on the second turn, since Crossbow Expert does not say the bonus hand crossbow needs to be different than the one-handed weapon that triggers its bonus attack. You may need to load it before the bonus attack using your object interaction, depending on your GM's interpretation of the feat. You can do this on each subsequent turn to fire a single hand crossbow twice per turn (or more than that, if you have Extra Attack).

You didn't request a rules-as-written answer, so I'll note that there is nothing game-breaking about allowing dual-wielding hand crossbows using Crossbow Expert.

Because of the way crossbows work, I think your GM would be perfectly justified in waiving the errata rule in this case but requiring you to get a customized crossbow and ammo belt to allow you to load it one-handed. That would be part of your house rules, though.