There have been many women who are the sole support for the family. At a woman's pay. Many times she is "the breadwinner" at 70/100 and she either works more hours, more jobs or makes do. And I believe it was a man/men who gave themselves the honor of "breadwinner".
Indeed there have, which is why pay should be (and should have been) much more equal. And I say "much more equal" only because two men doing exactly the same job don't always get exactly the same pay (experience and what-not comes into play), though it'll be fairly close.
As for men giving themselves 'the honour'...I don't know who started it - presumably it's as old as currency itself - but it would have been those in power and/or who had the most to gain who shut those doors. Other men would have found themselves in a position of being expected by society to become or remain as the provider (which stems from "man = hunter -> provider"). Over centuries that becomes embedded.
All I know is, I was raised in that way: men were the breadwinners, and a man who was not or could not adequately provide for his wife or family was a failure - and seen by society and other men as 'no man at all'. When you're raised to fulfil a role to the extent that it defines identity - just as women were/are - the fear of failure and all that it means can be a heavy burden...and one that the majority of men do not and did not ask for or (whisper it) even want.
Ultimately, though, if you have a system whereby men are given the breadwinner role or are expected to earn more than their wives or girlfriends, the situation is created whereby men have to earn more while women pick up relatively little. It never really worked in the past for the reason you point out, but those instances were in the minority and so nothing changed (leaving many families desperate as a result). Society has moved on and there are a great many more women who are sole providers, but there's still the idea about that men have to be the breadwinner - and that comes from women as well as, if not more than, other men, generally speaking.
It's not just pay equality that needs to happen, but full equality in terms of roles and expectations.
I always saw the "breadwinner" excuse for paying men more as just that, an excuse by employers to maintain the status quo--sort of an "old boys club" thing. I'd agree with you about there being a sharing of financial responsibilities in a household but that's always been my philosophy. These days, though, the majority of women who work outside the home are doing so because it's a financial necessity.
Yes. It shouldn't be - one income, no matter who earns it, should be enough to keep a household at least ticking along; anything else is simply unfair to single parents as well as singles without kids - but it is as it is for now.