The issue is not about bringing McCoy to Vulcan. The real issue is about bringing Spock from Genesis to Vulcan.
It's certainly a minor issue to bring McCoy to Vulcan. Rather, the point of much of the film is to bring Spock to Vulcan.
SAREK: Why did you leave him on Genesis! Spock trusted you. You denied him his future!...
KIRK: What must I do?
SAREK: You must bring them to Mount Selaya — on Vulcan. Only there is the passage possible. Only there can both find peace.
Retrieving Spock would be an extremely difficult feat, considering that Spock's body was laid to rest on Genesis and Genesis was now off-limits to all but a handful of Federation scientists.
Now, regarding Sarek's level of involvement, remember that Sarek was the Vulcan Ambassador to Earth and the Genesis planet had become a serious political issue. It is made clear in both The Search for Spock and The Voyage Home that the Genesis project was viewed as a weapon of mass destruction by the Klingon Empire, and the existence of the Genesis planet had become politically toxic, risking a flare-up of hostilities between the Federation and the Empire. Sarek, as one of the principal negotiators between the Federation and the Empire (as depicted in The Voyage Home), would have to stay clear of this.
- The issue is not about bringing McCoy to Vulcan (although that is part of the end goal). The main issue is about finding Spock, or Spock's body, and bringing both Spock and McCoy to Vulcan.
- Bringing Spock to Vulcan involves visiting the restricted Genesis planet, something that the principal Federation negotiator, Sarek, could not be involved with, even tangentially. Better leave it all to Kirk!