"Pushful" is a word I have never seen before, and seems
a clumsy, poor choice, particularly from Mencken.
I think "aggressive" would be my choice.

" It was reserved for Andrew Jackson, a man genuinely of the people, to lead and visualize the rise of the lower orders. Jackson, in his way, was the archetype of the new American—ignorant, pushful, impatient of restraint and precedent, an iconoclast, a Philistine, an Anglophobe in every fibre. "