Actually, this is interesting, because somehow everyone seem to know the rule for ח. In fact, I have a very clear memory of my rebbe teaching this rule to rows of us doe-eyed toddlers. Our אלף בינה had numerous examples to practice; מצמיח ,משיח etc. But nothing about the others. I can only assume that my rebbe didn't know it himself, which would not make him unusual. As a result, every month בעלי תפילה show if they do or do not know the rule as they publicly read בצאת ישראל ממצרים. It is especially funny when such prayer-leaders emphatically pronounce the
Obviously this is a symptom of the didkuk neglect of centuries (at least among Ashkenazim), often lamented, rarely rectified. The question is why this rule is only partially known. My guess is because of the three letters, only the ח is emphatically pronounced. Indeed, if Ashkenazim didn't pronounce the ח like the כ, but more like it's true Oriental consonantal sound partway between ה and ח, they'd probably have developed the pronunciation Moshihha!
In fairness to BMG, the name it's stuck with on its stationary and in it's legal filings don't reflect the dikduk knowledge or lack of on the part of anyone in particular. Although I'm not aware of any evidence that R. Aharon Kotler was a medakek in his pronunciation, certainly he knew the פתח גנובה. He had nothing to do with how the name is written in English; besides, if we're getting pedantic "Medrash" is misspelled too. Still, this is amusing, or at the very least, a perfect foil for my observation about why only a part of this rule seems widely known.