Crossword Genius

When a milk supplier is most essential to begin with? (7)

Ross

I believe the answer is:

teatime

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'begin with?' is the definition.
I can't tell whether this defines the answer.

'when a milk supplier is most essential to' is the wordplay.
'when' becomes 'teme' (I am not sure about this - if you are sure you should believe this answer much more).
'milk supplier' becomes 'tit'.
'is most' means to remove the last letter.
'essential to' is an insertion indicator (inserted letters contained by essential to others).
'a'+'tit'='atit'
'atit' with its final letter taken away is 'ati'.
'teme' enclosing 'ati' is 'TEATIME'.

Can you help me to learn more?

(Other definitions for teatime that I've seen before include "Four o'clock?" , "Afternoon meal period" , "When cake might be had" , "Afternoon meal break" , "Eat mite (anag.)" .)

I've seen this clue in The Times.
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