Proponent of a doctor drinking vodka ends with separation (9)

'proponent' is the definition.
(I know that proponent can be written as advocator)

'a doctor drinking vodka ends with separation' is the wordplay.
'drinking' indicates putting letters inside (inserted letters are consumed or drunk).
'ends' suggests the final letters.
'with' is an insertion indicator.
'separation' becomes 'v' (I can't justify this - if you can you should give a lot more credence to this answer).
The last letter of 'vodka' is 'a'.
'doctor' going around 'a' is 'docator'.
'docator' enclosing 'v' is 'dvocator'.
'a'+'dvocator'='ADVOCATOR'

'of' is the link.

Can you help me to learn more?