Some people in the early centuries definitely did associate the Old Man card with prudence after it lost the attributes of Time. The clearest example is Francesco Piscina's Discorso sopra l'ordine delle figure de Tarocchi
, from Piedmont in 1565:
il Vecchio Gobbo vien poi carco d' affanni e pensieri che vince e supera la Fortuna, il qual ci vol significar un prudente conseglio, col qual si vince ogni Fortuna [...] conciosiache un saggio e prudente superi il suo Fato e Destino, che par non so che di più d' humano, quanto maggiormente signoregiara e sarra vincitore della Fortuna, la quale e terrena, e per nulla  quasi da prudenti riputata, il Vecchio dico è posto per un considerato conseglio & ottimo giuditio, avvengha che sia proprio degl' huominii invecchiati e maturi il consigliar bene. Posciache accada che quel ch' essi dicono l' habino più volte per isperienza provato
Then the Old Hunchback comes, charged with troubles and thoughts, and he wins and surpasses Fortune. He represents a prudent counsel, with which you can win any Fortune [...] a wise and prudent man wins over his Fate and Destiny, for [in him there is] something that is more than human, as much as he is a winner of Fortune, which is earthly and considered to be almost nothing  by the prudent. I say the Old Man represents a well pondered counsel and an excellent judgement: it happens that it belongs to old and mature men to give good advice, because it is the case that they have experienced many times whatever matter they talk about.
[translation by Marco Ponzi, as revised in Ross S. Caldwell, Thierry Depaulis, and Marco Ponzi, Con gli occhi et con l'intelletto: Explaining the Tarot in Sixteenth Century Italy
(Lulu.com, 2018) ]
So yes, people did associate the virtue of prudence with the Old Man image. At the same time, I think it's fairly clear that people at this time (including Piscina) did not generally view the card as a personification
of Prudence, as such, and consequently they did not call the card Prudence, but instead always used names like Old Man or Hunchback.