I've only got better at hill climbing by constantly finding local hills to climb. It does work. Like most things. The more you do it, the better you become.
This is true, although there comes a point where no matter how hard you "train" (for hills or anything else) it's impossible to overcome physics and biology. Then there's that other limiting factor of one's psychological state. Most of us can't work as hard as we might, for all sorts of reasons, including the thought, "what's the point".
I know (because physics tells me) that if I lost one stone in weight, so I weighed what I did when I raced, I would go faster up the hills. I can hang on now to most of the club on the Sunday runs (the hardest of the week - I can't face the chain gangs) with only the skinny, younger, fitter lads getting ahead. So, do I need to keep up with the latter too? Not really. After all, I'm just cycling, not racing.
There is a strange meme in the heads of many British cyclists - that they must be "fast" or in some way "trying hard". This is a thing put about by the dominant cycling media in Britain but it's a thing which is rather silly really. And I say this as one who has succumbed! (It's the racing club traditions and habits, now built-in from my youth).
So, one answer to the OP is that alternative gears should not necessarily be to enable faster climbing but more comfortable climbing. And perhaps a kindness to the knee and hip joints.