Dang.
Why word CBE like that if the ammunition property ducks it up again?
Where is this line?"If you are holding one of them, you can load a piece of ammunition into it even if you lack a free hand."
No. CBE does two things.
It let's you ignore the loading property with this sentence:
"You ignore the loading property of the Hand crossbow, heavy crossbow and light crossbow (...)."
So I ignore the loading propert. Just like you said. The loading property restricts the amount of attacks per turn.
The next line of CBE says:
"If you are holding one of them, you can load a piece of ammunition into it even if you lack a free hand."
And that let's me ignore the part of the ammunition rules that says I need a free hand.
Seems clear to me.

(Dungeons & Dragons)
Rulebook featuring "high magic" options, including a host of new spells.