Little kiss welcomes a business man who's vain (7)

I believe the answer is:

peacock

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'man who's vain' is the definition.
I can't tell whether this defines the answer.

'little kiss welcomes a business' is the wordplay.
'little kiss' becomes 'peck' (I've seen this in another clue).
'welcomes' indicates putting letters inside (in sense of 'invites in').
'business' becomes 'co' (short for company).
'a'+'co'='aco'
'peck' going around 'aco' is 'PEACOCK'.

Can you help me to learn more?

(Other definitions for peacock that I've seen before include "Proud bird" , "Bird noted for showy plumage" , "Ornamental fowl" , "Proud strutter" , "Male bird" .)

I've seen this clue in The Telegraph.

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