I guess my take is that it depends on who is writing the reccomendation and what it actually says.
If it's truly someone with tremendous connections and power at the school, and the recommendation really says something like, "we must get this person" or "this person is once-in-a-lifetime" then I think that would be hard for adcoms to ignore.
If it's just some random alumni who doesn't really know you that well and gives a general recommendation, it probably won't help as much as an enthusiastic recommendation from someone that is a credible judge of talent.
I'm not sure what a "board of visitors" is, but if you worked closely with this person and he's enthusiastic about recommending you, it would be helpful.
From what I have been able to gather, recommendations trail GMAT, work experience & leadership/extracurriculars by a wide margin in terms of importance. I do not believe that even highly influential recommendations can overcome lackluster work experience or a low GMAT score. Now if you had $1,000,000 to donate to the school, that's another story...