I suggest the direction of influence is the other way round. Philosophical theories formalise and develop ideas that are already in circulation. Remember Hegel's observation that 'the owl of Minerva spreads its wings only with the falling of the dusk' : philosophical theory comes afterwards, reflectively, when a development of ideas or institutions is complete and (he would add) in decline.
Plato's 'Republic', at least its political portion, was as the late Michael Oakeshott once put it, 'animated by the errors of Athenian democracy'. Any citizen could participate in politics and help determine policies and legislation without any knowledge of the relevant matters. Plato saw democracy as the politics of ignorance. If every other human inquiry or activity recognised expert knowledge - in his famous example, you wouldn't let just anyone, regardless of their lack of specialist skills, navigate a ship - why not politics, too ? Why should politics be special in not requiring knowledge of the proper ends and means of political action as a condition of participation. Think of this what you will, but the 'Republic' was rooted in its contemporary context and was a response to it.
Aristotle's 'Politics' is a theorisation of the Greek polis, which was already passing out of independent existence under the impact of Alexander the Great's conquests.
Aquinas' 'Summa' was a response to the recovery of Aristotle's writings and to the ongoing beliefs and practice of the Catholic Church - as well, of course, to movements which he opposed in theology.
Hobbes' 'Leviathan' is clearly a recipe for avoiding the kind of political and social chaos caused by the French Wars of Religion and the English Civil Wars. (They were in his rear-view mirror.)
Hume's 'atomistic' view of the nature of experience as composed of distinct impressions and ideas drew on the model of Newtonian 'corpuscular' physics.
Kant's Critique of Pure Reason asks how knowledge is possible, with the glories of Newtonian physics in the background. His emphasis on the place of reason in ethics is fully in the spirit of the Enlightenment's celebration of reason.
John Stuart Mill's 'On Liberty' was a counterblast to the pressure toward conformity which he thought he saw in the England of his day.
Logical Positivism was a response to the huge, brilliant developments in science - relativity and quantum theory - and took the form of scientism, the view that scientific knowledge is the only form of deep and accurate knowledge (of all real knowledge).
I could go on. I am not suggesting, of course, that there was anything crude or mechanical in the way these philosophies emerged from their contexts. They all added independent thought of great subtlety. But their problems and the terms of their solutions were set by their times, at least as they understood them.