I doubt that saying a word with a slight variation in meaning is a misstate. Like genetic evolution, things can just change. Small changes need not be viewed as mistakes. I have no problem with what "nice" means today regardless of what it meant originally. This is missing the point. "Forget" is a technical issue. "Forget" evolving to mean "can't remember," did not eliminate its original meaning of losing grasp of the information. We still have that meaning. If "nice" still meant "stupid," as well as its newer meaning of "kind," there might very well be a problem.
"Forget," as stative, has a bigger problem though. As, mentioned previously, one cannot have an act and a state contained within the exact same word having the same general meaning. "Think" can have both aspects, but each of those two aspects have a different meaning: "I think Lady Gaga is a great singer," and "I'm thinking about her now." Besides having two different meaning, the dynamic aspect is distinguished by adding -ing. There is no parallel to the usage of "forget" having two aspects in the same word, meaning the same thing, in the same tense. This is contrived and must be viewed as ungrammatical.
There is also no reason to view opposites as being mirror parallels, or having "symmetry" in concepts, and in how they can be treated. In fact, opposites may always need to be treated differently, at least with verbs: "I have a pencil." / "I don't have a pencil." These certainly sound opposite, but they are not. In "I have a pencil," one had to be given a pencil, but to not have a pencil, one did not need to lose a pencil. There is no mirror image. One can only say "I don't have a pencil," or "I lost my pencil." Likewise, with "remember," one can only say "I don't remember," or I forgot" (I lost the information just as I lost my pencil.) Saying, "I forget," to mean "I don't remember," is a non sequitur. There is no possible word that can mean not remember because there is no opposite state.
No opposite state? That's correct. When one says, "I can't remember," one is actually saying he/she cannot get back into the state of "remembering." That is only a declaration of not being able to get back into the previous state. I can drive a car. - (state) / I can't drive a car - (not in the state of being able to drive a car). What verb could mean a state of not being able to drive a car? One can only say, "I can't drive a car." "I have a book." "I don't have a book because I lost the only one I had." Lost is the reason why the person doesn't have a book. That cannot be turned into a state of "I lose my book." Perhaps a word to mean not having the book could be "bookless," but that would be an adjective, like not having a home is homeless. There is no verb-state that can describe not having the book.
So now, let's look at your examples:
"When I was gardening yesterday, I remembered his name." is the parallel to "When I was learning German, I forgot all my Greek".
= I became (un)able to recall X. (cf. meaning #1)"I remember that day clearly" is the parallel to "George or Bill, I forget which".
= I do (not) retain the information (cf. meaning #2)
First set:
In the first sentence you got information / lost information / and got it back. (Logical and Grammatical)
In the second sentence you got information / then lost it.
Not perfectly parallel, but both are dynamic; you remembered, and forgot. No logical problems here. Your word "forgot" just means you lost the information you had gotten at some previous time. (Logical and Grammatical)
Second set:
In the first sentence you retained information from a previous experience. (stative) There was never an act where you had to have "remembered" anything. So, there is nothing going on dynamic here at all. The word "remember" need only represent retention of an experience. (Logical and Grammatical)
In the second sentence you got information / lost the information /can't get the information back.
There is no parallel here at all. Your word "forget" must somehow mean got - lose - can't get back. Impossible. It makes no sense; no stative verb can represent lost the information you previously had gotten, and now can't get that information back. One must say, "... I forgot which." Since you can only lose something you first got, both the got part and the lose part are covered. Since "forgot" is the reason for "can't get the information back" you're covered there as well.
Now using your own logic against you. You have said one should not say, "I forget my 11 times tables." So, why not just argue that forget means the opposite of "remember" here too, as you did above? How can it make sense in one, but not the other?
Grammar is the logic of how words are put together. It's not about usage. The sentence "A or B, I forget which," is illogical, thus ungrammatical, as is any stative usage of the verb "forget."
