It's a case-by-case thing.
I doubt people can name the six #1s that Paula Abdul had, and a good bit of people probably don't know that "Heartbreaker" or "Thank God I Found You" by Mariah Carey were #1 singles.
Some of the most acclaimed acts have gone on to be legendary - The Beatles, Aretha Franklin, Marvin Gaye, The Rolling Stones, etc. But then there are a ton of critically acclaimed artists that don't have that luxury and are forgotten completely.
And while people may not know that Sir Georg Solti (I've never heard of him, so this is obviously me playing the devil's advocate) is the most awarded artist in Grammy history, many people are probably aware that Beyonce, Kanye, Aretha Franklin and several artists are Grammy winners.
There's no one way an artist can go down in history.