I think it is definitely the case that most physicists have quite a wrong picture of past physics.
I have some historical information about the work of Joseph Louis Lagrange. (I don't think it affects the thrust of your post.)
An english translation of Lagrange's 'Mecanique Analytique' is available on Archive.org. Analytic mechanics Lagrange used calculus of variations for problems in statics, but not for problems in mechanics.
At the time the action concept that was available was Maupertuis' action. (Hamilton's action was introduced by William Rowan Hamiltons in 1834, Lagrange died in 1813) In his work 'Mecanique Analytique' Lagrange offers the opinion that Maupertuis' action is not particularly relevant. Quoting one sentence from page 183 of the english translation. " [...] which I view not as a metaphysical principle but as a simple and general result of the laws of mechanics."
I want to emphasize the necessity of using the name 'stationary action'. You do acknowledge that action can be "either minimized or maximized". To acknowledge that, and to keep using the name 'least action' amounts to self-contradiction. I understand that many people prefer the name 'least action' because that just sounds sexier than 'stationary action'. But it is what it is: minimum or maximum is immaterial. The criterion is: identify the point in variation space such that the derivative of Hamilton's action is zero.