Pigeon in sink gobbling insect (7)

I believe the answer is:
fantail
'pigeon' is the definition.
(type of pigeon)
'sink gobbling insect' is the wordplay.
'sink' becomes 'fail' (I am not sure about this - if you are sure you should believe this answer much more).
'gobbling' indicates putting letters inside.
'insect' becomes 'ant' (ant is a kind of insect).
'fail' going around 'ant' is 'FANTAIL'.
'in' acts as a link.
Can you help me to learn more?
(Other definitions for fantail that I've seen before include "Type of domestic pigeon" , "Flier" , "In a flat (anag.)" , "Bird" , "Variety of pigeon" .)
