The most common and most likely answer to this crossword clue is the 6 plus 5 letter word MISTER RIGHT
I've seen this crossword clue in Financial Times
M
I
S
T
E
R
R
I
G
H
T
Buddy explains!
The straight part
We know that "Man who wouldn’t be left on the shelf?" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case man who wouldn’t be left on the shelf? = mister right.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.