You asked some very perceptive questions. Let me answer each one of them in some detail:
1. I’m afraid that the distinction between “healthy” and “healthful” has been—and continuous to be—eroded in contemporary usage. This is largely because of the propensity of many food manufacturers to use the adjective “healthy” instead of the scrupulously correct “healthful” in advertising pitches for their products. By definition, “healthy” means “enjoying health and vigor of body, mind, or spirit” and “evincing health,” but it also has the third meaning of “conducive to health.” This third meaning is, of course, practically synonymous with that of “healthful,” whose primary meaning is “beneficial to health of mind and body.” Food manufacturers and their advertising agencies are therefore not committing prevarication or deviating from truth in advertising when they prefer to use “healthy” over “healthful” in their ads or product labels. They are simply exercising their freedom of word choice. Evidently, from a marketing and creative standpoint, they obviously find “healthy” a more compelling pitch for their food products than “healthful.”
2. Yes, I did say in my book English Plain and Simple that the modal auxiliaries “should,” “must,” and “ought to” could not work with “not” to form the negative of their affirmative statements, but I specified at the outset that this is only in the context of their function of indicating varying degrees of obligation, necessity, or certainty. These three modal auxiliaries actually have many more meanings and usages than the one contemplated by the discussions in that chapter, and the sentences you provided as examples reflect these other meanings.
For instance, in the statement “But this aid should not be seen as a blank check,” “should not” is being used in the sense of a prohibition or exception, not in the sense of an obligation, necessity, or certainty. This is in contrast to one of the usages of “should” that I provided using this latter sense, “You should pick up your sister from the airport,” which is a clear case of an obligation or necessity. When that statement is made in the negative sense by adding “not” to “should,” as in “You should not pick up your sister from the airport,” it’s no longer a statement asserting obligation or necessity on the part of the person being addressed but a prohibition; in fact, this negative statement has a sense of semantic incompleteness or lack of resolution—one that can be resolved only by adding further information like, say, where the sister should be picked up instead, as in “You should not pick up your sister from the airport but from the house of your cousin Rico instead where she’s staying overnight.” (This sentence is actually a loose, colloquial construction of this more scrupulously correct construction, “You should pick up your sister not from the airport but from the house of your cousin Rico instead where she’s staying overnight”—a telltale indication that there’s really something grammatically amiss in putting the words “should” and “not” adjacent to each other in sentences where “should” is meant in the sense of an obligation or necessity.)
This was my point when I said that “should” could not work with “not” to form the negative in such grammatical situations, for “not” cancels or contravenes the sense of obligation or necessity and turns the statement into a prohibition. The same thing would happen if we make “must” and “ought to” work with “not” to yield a negative sense, as in the other sentences you provided as examples. The intended sense of obligation or necessity will be cancelled in the statement and is replaced with that of a prohibition or an exception.
3. In my book Give Your English the Winning Edge, I did give this example of a sentence with an adjective clause: “The strategy which they used to win the bidding was superb.” This is an example of how British English normally uses the relative pronoun “which” to introduce a restrictive clause without setting off that clause with a pair of commas, in contrast to American English, which uses “that” to do the same thing, also without setting off the clause with a pair of commas. You are correct, of course, in observing that in American English, the relative pronoun “which” is normally used only for introducing nonrestrictive clauses that, in turn, must be set off by a pair of commas from the rest of the sentence. In the context of the discussions in that particular chapter of my book, however, I used the sentence in question as a random example of actual grammatically correct English sentences—regardless of whether they are in American English or in British English—that can be reduced by eliminating their relative pronouns without damage to their grammar or intended meaning. In Part V of the book (“Matters of Usage”), I devote four chapters to discuss and clarify the nuances of the usage of “that” and “which” in introducing restrictive and nonrestrictive clauses, and show how these relative pronouns may be eliminated to reduce those modifying clauses into grammatically simpler adjective phrases.
