It is proper military task? Of course it is a proper military task - MT2.1 Military Aid to the Civil Authorities and, arguably, MT2.3 Integrity of UK Waters. MT4.1 Humanitarian Aid and Disaster Relief might be stretching it a bit far, as that is overseas contingency tasking.
Why should Defence provide this service? The MOD has thus far kept its own SAR services in order to recover its own valuable assets in the event of accident. The requirement endures but the number of military callouts is few, so the irreducable spare capacity is used to aid the general public (people get saved + SAR crews get real on-the-job training = everybody wins) on the understanding that the military gets first refusal should the need to prioritise arise. Imagine the (entirely justified) outcry if military SAR assets were sat on the ground when a civil need arose because the one GR4 airborne at the time might need support.
The vast majority of SAR's 'customers' are civilians. The amount of military activity in the UK is now so low that the MOD can possibly take the risk of handing the task over to the DOT, accepting that we are no longer prime inter pares and will have to wait our turn should a military requirement coincide with a civil one. Someone will have done the sums and decided that the risk of relying on a civil SAR service is acceptable in terms of response time and availability balanced against the through life cost of procurring and operating a replacement for SeaKing.