It was famous (insofar as any work of art was famous) from the get-go. Giorgio Vasari (1511-1574), in his Lives of teh Artists, writes as follows:

"Leonardo undertook to paint for Francesco del Giocondo a portrait of Mona Lisa his wife, but having spent four years upon it, left it unfinished. This work now belongs to King Francis of France, and whoever wishes to see how art can imitate nature may learn from this head. Mona Lisa being most beautiful, he used, while he was painting her, to have men to sing and play to her and buffoons to amuse her, to take away that look of melancholy which is so often seen in portraits; and in this of Leonardo's there is a peaceful smile more divine than human."

Giorgio Vasari: "Life of Leonardo da Vinci", in Lives of the Most Eminent Painters, Sculptors, and Architects.

He discusses any number of works by people he knew, or nearly knew, many examples of which will be familiar to this day. He was, or course, writing before serious wide scale reproductions of any quality were widely available. At best, one might find a wood cut taken from a known painting (there are a few cases where these copies are all we have, the originals having been lost or destroyed).

Then there's the question of market forces. A name artists working for moneyed patrons could do quite well in the sixteenth and seventeenth century. Connect high prices with the artists getting paid and the ease of access to the final product by the man in the street and you get an idea of what worked. (It was easier to go to a church and see a Titian than to go to Paris and ask the king if you might take a peek at la Gioconda).

But to your question - There have been paintings that had their moments in the sun, but for your purposes, and not excluding the possibility of others, I would nominate Il Moro (1650) by Velasquez. The painting drew long line and crowds from the moment it was unveiled in the Pantheon in Rome, and continued to stun fellow artists when it hung in the Prado in Madrid. (Edward Manet had nice things to say about it in 1865.) Thomas Hoving, then director of the Metropolitan Museum was considered almost reckless when in 1970 he bought it for a record one million pounds., but it's proven a good value. It's still there and well worth a visit if you're into world class portrait paintings

There are no doubt others that reach the Platonic ideal you're asking about, but that's what comes to my mind off hand. I hope others chime in.

Spear, R, Painting for Profit: The Economic Lives of Seventeenth-Century Italian Painters, Yale University Press