Of course there were many influential painters in between, but if you Google a list of famous painters, the names will disproportionately come from the Renaissance and Modern eras. So that's my standard for "famous."
This question is more historiographical than historical. When we talk about classical European portraits, we're thinking of a movement that began with the Renaissance, and produced an explosion of well-known individuals that basically ends in the early 17th century with Rembrandt, Velazquez, and Caravaggio. There are a few famous ones in the early 19th century (JLD, Turner, Constable, Friedrich), but there's another explosion at the end of the 19th century with Impressionism and Cubism. Here's a list of 101 famous painters. I didn't count, but it seems that there are far more painters from 1870-1910 than the previous 200 years combined. Why is that?
One answer would be that these painters created modern art, and because we continue to live in the era of modern art, they're held in high esteem. This makes sense. But it doesn't explain why we have held on to Renaissance painters more than the painters from 1650-1800. Why are 18th century painters so overlooked?
I can't offer more than general comments on the Modern Era, but given that my specialty is Renaissance I can provide some context there.
Probably the biggest factor for the popularity and sway of Renaissance (and in particular, Italian Renaissance) artists is Giorgio Vasari, the 16th-century author (and artist himself) who wrote Le vite de' più eccellenti pittori, scultori, e architettori (colloquially known in English as “The Lives of the Artists”). Published initially in 1550, and then expanded and released in a subsequent edition in 1568, “The Lives” is easily one of the most important art historical texts of the Western (i.e., European) tradition. I don’t think it’s an overstatement to say that we are, to some extent, still living within the shadow of Vasari and the Vasarian mode of art history.
“The Lives” are, at heart, a series of biographies of famous painters, sculptors, and architects, starting with Cimabue in the 13th century and going up to contemporary artists like Titian (still alive at the time of his writing).
Beyond that, it is also a history of the visual arts and an attempt to provide a big-H “History” of the rise, fall, and rebirth of the arts, from antiquity up to the author’s own day. The culmination of this story (for Vasari) is Michelangelo. In him, Vasari saw an artist who not only equaled the ancients (i.e., the great artist of Ancient Greece and Rome) but who exceeded them.
And it wasn’t only Vasari who felt this way—in many 16th-century circles, this was the accepted consensus. As Patricia Emison has noted:
“By the time of the death of Michelangelo […] there was only the possibility of comparison to Michelangelo, who was known as learned, in anatomy if not necessarily in geometry; as a poet in his own right and a commentator on Dante; and as the most famous of Florentines, recipient of the most lavish citizen funeral in memory, accorded the most extravagant tomb in Santa Croce, and the subject of not one biography but two.” (Emison 2004, 59)
Vasari was instrumental in elevating and memorializing Michelangelo as not just an artist but as a semi-divine figure whose fame was international (in Europe, at least). While there had been writings about artists prior to Vasari, none were as thorough—or as thoroughly researched—as Vasari’s “The Lives.”
Admittedly, Vasari gets a lot wrong, and in certain areas he’s doing a lot of myth-making (to nothing of outright fabrication), but he also seems to have worked hard to try to get accurate stories.
All this is important because it means he set the template for a lot of art historical writing: namely, a focus on “progress” in the arts (remember: rise, fall, rebirth), a focus on the individual (“Great Men” of history, although Vasari does include a few women artists), and an emphasis on Antiquity and Nature as the markers of quality. By providing this template and these yardsticks by which to measure artists Vasari all but guaranteed that subsequent histories of art—for a long while—would either be refinements of his approach or reactions against it.
And indeed we see that some of the art historical texts to come out in the wake of Vasari are basically efforts to add to his canon in a mode that is “Vasarian”—e.g., Karel van Mander’s Schilder-boeck, which is a 17th-century Dutch history of famous artists that expands on (and was heavily influenced by) Vasari’s work.
The wealth of information provided by Vasari, and the fact that he so firmly established the era immediately preceding his own as the “pinnacle” of artistic achievement (embodied in Michelangelo), meant subsequent generations were almost always going to be predisposed to the Renaissance and its artists.
I’m happy to have others chime in here, but I don’t think there’s a comparable art historical text—in breadth, reach, and influence—in the Western tradition, full stop.
As for why Modern Art is similarly crowded or over-represented, I will leave that to others.
sources:
Patricia A. Emison, Creating the “Divine” Artist: From Dante to Michelangelo (Leiden: Brill, 2004)
Vasari (of course), who is available in any number of modern editions and translations (including some free online)