it does say that. prayers made for all men, God wants all to be saved.
i have heard the all kinds of men or all classes of men argument by james white, but i think its very weak and you have to stretch it unnaturally to make it say that. like titus 2:11.
calvinists make this in all the spots where it says all men, because it doesnt fit L of tulip, so they have to invent some way out of it. i can imagine free will believers also have some verses they have to invent a way out of, like romans 9. i think free willers explanation are really weak when you just read romans 9 as is, they end up agreeing with the objector. but i also believe the same about calvinists reading of titus 2:11 and 1 timothy 2:3-4, too unnatural reading.
there must is a way to harmonize all scripture?