Broilers aren’t kept in cages generally, but if you don’t keep layers in cages then it’s a lot more labor to collect the eggs and make sure they don’t just eat them or break them. So the lowest quality eggs will come from chickens that live in cages stacked several rows high, with an incline in the bottom of each cage, so that when they lay the egg will roll onto a sort of conveyer belt that moves the eggs over to be packaged.

Source: my rural ass high school had ag classes and we went to some of these places. I guess it’s possible this has changed in the past 20ish years, but from what I know it hasn’t changed that much. If you didn’t grow up down wind of some of these places, consider yourself lucky.

All the chickens get transported to the slaughterhouse in cages

Ok, but that's not what cage-free means, cage-free means they don't live in a cage. How they're transported was never part of it. I'm guessing "free range" chickens are transported in cages as well, because that's a lot easier.

They’re raised in a chicken house

Idk, this looks like a cage to me.

equivalent is if you grew up with a 60sqft personal bubble

That doesn't make sense, humans stand upright, chickens are more long. I'm guessing the size comparison you're talking about is total size, not size on the horizontal plane. A chicken is something like a foot long and a half foot wide, or something like 1.5sq ft. That meas there's half their body length in space not filled by another chicken in a 2sq ft area. That's not a lot of room.

chickens are a family pack animal, so even if they have all the space in the world they won’t use it

Chickens are foragers, so yes, they'll absolutely use the space provided. I have friends who raise chickens, and live in an area where raising chickens is common, so I know what chickens do. If I ever forget, I can walk down the street and watch chickens for a half hour and see what they do. They don't clump together, they spread out to forage for bugs and whatnot, and they only pack together when they go back to the coop to sleep, or if they are in danger (there is safety in numbers).

But yes, chickens are quite dumb.

modern broilers cannot live long just due to their genetic breed

Well yeah, they're genetically selected to have maximum meat because that's the most efficient way to farm chickens.

Likewise for egg-laying chickens, they're selected for volume and consistency of egg output. Some breeds make brown eggs, some make white eggs, and those are sold to different markets (usually brown eggs are sold at a premium here because people think they're better in some way; they aren't).

The two types of chickens (eggs and meat) are generally not the same, and my understanding is that egg-laying chickens are discarded rather than sold once they stop laying. I could be wrong (maybe they're used for chicken nuggets and other processed chicken products), but they're definitely not used for the cuts sold at stores because their meat is too tough.

If it’s really something that bothers you

It's really not, I'm merely pointing out what the terms mean. I buy whatever is cheapest at Costco (currently cage-free is the lowest tier), and if I cared about the welfare of the chickens, I'd raise my own (and their eggs taste better anyway).