The RCAF doesn't want to yield aviation turf to other players. From a cost point of view, duplicating aviation recruiting, training, and acquisition of kit across several departments would suck. Neither the RCN or CCG want the added expense of supporting relatively small helicopter fleets from their already lean budgets.
WRT SAR, I seriously doubt SAR is 1/3 of the RCAF's budget. Assigning SAR to the private sector would require huge subsidies to entice companies to service remote regions. They and Transport Canada would have the same issues the CCG and RCN in supporting small aviation fleets. Canada has treaty obligations wrt SAR in the Arctic and frankly our current and planned acquisitions don't really provide enough capability. An airliner ditching or cruise ship disaster in the high Canadian Arctic is a place we wouldn't want to be at.