Old player in cricket team backed by leader of Conservatives — would one often open for Churchill? (5,3)

'would one often open for churchill?' is the definition.
(Winston Churchill smoked cigars)

'old player in cricket team backed' is the wordplay.
'old player' becomes 'Garbo' (actress Greta Garbo, c905-c990).
'in' indicates putting letters inside.
'cricket' becomes 'c' (this could be a standard abbreviation I've not previously seen).
'team' becomes 'XI' (a football team, from Roman numerals for cc).
'backed' is a reversal indicator.
'xi' backwards is 'ix'.
'c'+'ix'='cix'
'garbo' placed inside 'cix' is 'CIGAR BOX'.

'by leader of conservatives' is the link.
I am not very happy about this link. Some or all of it may belong to another bit of the clue.