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SilentCal comments on Does utilitarianism "require" extreme self sacrifice? If not why do people commonly say it does? - Less Wrong Discussion
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My view, and a lot of other people here seem to also be getting at this, is that the demandingness objection comes from a misuse of utilitarianism. People want their morality to label things 'permissible' and 'impermissible', and utilitarianism doesn't natively do that. That is, we want boolean-valued morality. The trouble is, Bentham went and gave us a real-valued one. The most common way to get a bool out of that is to label the maximum 'true' and everything else false, but that doesn't give a realistically human-followable result. Some philosophers have worked on 'satisficing consequentialism', which is a project to design a better real-to-bool conversion, but I think the correct answer is to learn to use real-valued morality.
There's some oversimplification above (I suspect people have always understood non-boolean morality in some cases), but I think it captures the essential problem.
A useful word here is "supererogation", but this still implies that there's a baseline level of duty, which itself implies that it's possible even in principle to calculate a baseline level of duty.
There may be cultural reasons for the absence of the concept: some Catholics have said that Protestantism did away with supererogation entirely. My impression is that that's a one-line summary of something much more complex (though possibly with potential toward the realization of the one-line summary), but I don't know much about it.
The word may have fallen out of favor, but I think the concept of "good, but not required" is alive and well in almost all folk morality. It's troublesome for (non-divine-command) philosophical approaches because you have to justify the line between 'obligation' and 'supererogation' somehow. I suspect the concept might sort of map onto a contractarian approach by defining 'obligatory' as 'society should sanction you for not doing it' and 'supererogatory' as 'good but not obligatory', though that raises as many questions as it answers.