If I recall the whole thing was deeply personal and embarrassing, and with Sareks powerful Vulcan emotions escaping his control he was even more stubborn in refusing to admit a problem or that he needed help.
Picard, being an accomplished diplomat himself, was able to convince Sarek he was putting the mission at risk by not accepting help. And he was a good fit for the mind meld both because of his ability and because he was a human. Because let’s face it, most vulcans are judgmental as heck (which probably would have bled through a mind meld)