If a global UN Search and Rescue organisation were considered unpalatable, might REGIONAL SAR Coordination Centres, each with the authority to control assets from member's forces/emergency services be a better (less sensitive) option than an all-out UN umbrella organisation?
There is a international Maritime SAR document but indeed loosely worded. More importantly, ICAO does spell out the authority and responsibility for SAR in maritime disasters.
The countries here Malaysia, Singapore, Thailand, Indonesia, Australia, US, China (source: The New Straits Times, not sure why they left out Vietnam), had worked closely before in such or similar situations (Boxing Day Tsunami). Perhaps only China is the 'new player' but a good 85% do have past experiences. Sure, people rotate out but certain SOPs do get passed down. Additionally, most of these countries have conducted rescue exercises together, one of this is a submarine rescue exercise (not sure of it's frequency).
So to reply to your suggestion, there are sufficient multi-lateral / coalition / UN tasks out there for the Armed forces of modern countries to work together. The framework to cooperate is usually very standard and there isn't a dire need for more organisations. Hope this helps.