The USMC has decided that it cannot devote assets exclusively to CSAR. We've decided we can't accommodate buying dedicated equipment nor dedicated personnel, either in aircrew or PJ equivalents.
Instead, it has decided that TRAP is the best way to take care of its responsibilities to recover aircrew in a combat environment. Of course, in the purple (joint) world we work in, there is a division of responsibilities between the services whenever there is a major operation going on. There's actually a joint agency that steps in to coordinate which service provides what when they're all at the party.
When the Marine Corps is working on its own or off of shipping there's a matrix used to determine what assets go get a downed crew, depending on threat and distance from friendly forces. Generally speaking, overwater the Navy helos will get you. Overland, various TRAP packages are assembled depending on the threat.
Station SAR is important, but because there's no enemy, it doesn't matter so much WHO does it, just that it gets done. Hence the reduction in station assets in favor of outside agencies. Iwakuni got rid of its SAR in favor of Japanese assets, and Beaufort SAR got replaced by the USCG. Again, prioritization based on risk/benefit analysis takes place, e.g. helo bases don't get SAR.
The Marine Corps will help out the civilian side when asked, but that really can't be its focus of effort. As far as the Osprey, the Corps is plenty busy getting squadrons ready to deploy, without worrying about training for extra missions.