I really can't figure out where the masu form comes from. I thought it might
be from kobun volitional む + causative す but that's -(a)ます not -(i)ます.

Both conjugations of ません and ましょう are identical to how the Kyoto dialect
conjugates す. That's as much as I could figure out, I'm really stumped here.
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grammar etymology classical-japanese
Where does the ます形 masu form comes from? I noticed that the す conjugates like the Kyoto dialect
According to Frellesvig ( _A History of the Japanese Language_ , p338), the
_masu_ auxiliary is derived from the polite auxiliary _marase_ , which
apparently is a humble word for 'give'. The _marase_ form emerged in Late
Middle Japanese (1200-1600) and is itself a causative form derived from the
earlier _mawir_ ('come').

Beyond that, I don't know anything, but I suppose you can imagine the concept
of 'giving' being applied to one's own actions to suggest a
humbleness/politeness in a similar way that the current ～てあげる is used.
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