Seems like an apt description to me. If you're referring to killing chickens for consumption, eggs have no bearing, it has its own demand.
There's no way to sell eggs and turn a profit without killing hens. Their production wanes as they get older and they are killed to make way for the new generation. This has nothing to do with demand for meat. Your parent comment is right; these are not "no-kill". At best, they're "less-kill".
I believe this is true of conventional caged farming, not all of it. If they can aspire to "no-kill" on one level, no reason they can't do so across the board.
> There's no way to sell eggs and turn a profit without killing hens.