Not backing him up, wouldn't do for the chairman (5)
I believe the answer is:
stool
'not backing him up wouldn't do for the chairman' is the definition.
Although both the answer and definition are singular nouns, I can't see how they can define each other.
'him up wouldn't do for the chairman' is the wordplay.
I cannot quite understand how this works, but
'for' could be 'to' (eg both can mean 'in the opinion of') and 'to' is present in the answer.
'him' could be 'sol' and 'sol' is found in the leftover letters.
This accounts for all the letters.
This may be the basis of the clue (or it may be nonsense).
Can you help me to learn more?
(Other definitions for stool that I've seen before include "Seat without a back" , "Simple seat" , "Small seat" , "Backless chair" , "item of furniture here" .)