You've got normal cage which seems to typically be useful for picking up small animals but can be configured to allow others and you've got the mechanic to allow mobs to only be picked up on a health threshold.
While being able to pick up all things is not survival friendly (as far as the config thinks) would it be a possible idea to expand upon this and allow 2-3 forms of cage?

The normal cage which can pick up small animals and mobs or baby animals and mobs without a limitation.
But also a "powerful" cage made with 3 normal cages as the bars and obsidian along it's bottom with a diamond and two gold on the top that produces a cage that renders much larger (would take up the space of maybe a 2x2x2 when placed on a block and the entity would render inside up to a certain size) which can capture bigger mob types.

Mobs who are large but passive/neutral or weakened large hostiles tagged as "cageable_large" or something of the sort. But then also mobs who are tagged as "cageable_powerful", basically for bosses and other powerful enemies who must have their health reduced by an amount to be viable to capture.

Alternatively, smithing tabling a nether star onto a large cage can make it a master cage giving it a more crystaline and enchanted glow which now takes up 3x3x3, which has it's own configurable HP reduction amount which would be intended as a super capture for most if not all things. (or can be made to have a higher threshold for less frustration/straight up allow any capture)

Then you can let people have their small cage for original functionality. Large cage for the mobs and potentially bosses if configured and with the struggle, then the master enchanted cage for the "go nuts, catch and display overpowered things" tiered functionality for when they're approaching their end game. (depending on what their end game is, people can of course just edit the recipe with datapack after the fact but the functionality would atleast be there)