I've heard "gentlefolk" used in communities where outright "cultural re-write" for less sexism and racism is tolerated or encouraged (rather than inducing apoplectic sputtering by some of the reactionary participants). Lesswrong, with the semi-regular and un-commented use of spivak, is one of these places.

On the other hand, to my ear, the penumbra of connotation included by "gentlemen's agreement" includes elements of feudal virtue and the glamor of evil, where gentlemen on hard times might become highway brigands rather than shop keepers or farmers, because it would be ignoble to work for a living rather than finding some way to prey on the productive members of society "like a proper gentleman".

Legitimate prey, in that era, only vaguely includes women, because women were mostly just "points" rather than players, formally existing as willing or unwilling chattel, rather than official players of the game. The residuum of this era seems to be what relatively sane feminists are talking about when they speak of "the patriarchy" and "rape culture".

There were exceptional women, but "gentlefolk" doesn't make me think of barely restrained rampaging female nobility, the way "gentlemen" subsumes a historical panoply of somewhat foppish, honor obsessed, sociopathic dukes and princes, who had "always winning at ultimatum games and games of chicken" as the essential business model of their caste.

The caste, to persist over time, needed a way to not tear itself completely to shreds through internal squabbling and so develops cultural norms regulating internal strife... hence, a "gentleman's agreement" could occur between its genuine members. Figuring out a way to identify genuine members (by some method other than never backing down when a peasant tries to get an even deal) loops you back to signaling, completing the circle of game theoretic horror.

In the meantime, yay for progress, bringing humans slowly out of the bad old days, one semi-horrible half-cynical compromise at a time :-)

With the terminological question, I think its worth comparing a "feudal horror" understanding of "gentlemen" with someone like Jeanette Rankin, the first female ever to be elected to the US House of Representatives, who was a pacifist and the lone vote against declaring war on the Japanese after Pearl Harbor. The word might have no cognate because it is inherently gendered? I think it's distinctly possible that there are few female gentlemen for basically the same reason that there are few female serial killers.