The MEL is a "dispatch" document that applies "...until the throttles are advanced for the purpose of taking off ( the takeoff event )". Once airborne, the POM ruled, but the MEL was always consulted as another source of info/ideas/planning. I can tell you how it was done at Delta from 1973 until I left in '03.
To depart with an MEL item involving anti-ice equipment ( window heat inop, bad TAI valve, etc. ) required a " no ice clearance" from the dispatcher per MEL. This was not a decision one, as captain, could make even if he wanted to. It involved meterologists and the dispatcher and was a legal parameter. Of course, one could always just refuse to fly inspite of the no-ice clearance because he felt it would be unsafe to fly...that's always the captain's call.
If the equipment in question went inop enroute, there were POM procedures in insure adequate anti-icing to complete the flight. Again, if the captain didn't want to procede to destination due to the nature of the problem ( maybe ALL wing, or ALL tail ice protection was inop as an example ), it was then HIS call and would not be questioned by DL Flt Ops management...I can assure you, unless DL has really changed in the last year or so.
Did this address your original question ? Without a lot more details, this is the best I can offer.
The legalities are a different matter from the parameters of temp and viz moisture which dictate WHEN one turns on engine TAI, for example.