On the whole, the article's summary of the situation is accurate. Since the 1960s or so, explicit grammatical teaching has been denigrated by many in the teaching establishment, and so now, a generation of teachers have gone into service without any understanding themselves of what this grammar stuff is.
The article's hook is a teacher, one Mike Greiner, who insists on doing things the old way. This strikes me as a predictable, and wholly unfortunate way of coping with the SAT. Throwing out explicit grammatical instruction entirely was a tragedy, but the old ways of teaching grammar were lacking as well.
The books used were (and still are) filled with factual inaccuracies and misleading inferences. Teaching proceeded by drill-and-kill, context free exercises, and what was called grammar in fact was a jumble of unconnected points, many of which are entirely trivial.
Take a look at Mike "Grammar" Greiner's Seven Grammatical Sins, presented in a side-bar to the piece:
7. Griener. It's Greiner; I'm an exception to the rule about i and e.
6. Many writers think commas are cool, semicolons are special and sophisticated. Use a semicolon between independent clauses.
5. english. Capitalize the most important subject . . . or proper nouns.
4. Between you and I . . . Use the objective case, i.e., me, after prepositions.
3. This is her. Use the nominative/subjective case for predicate nominatives.
2. Grammer. If you'd like an A in English, spell grammar with 2 A's.
1. I'm doing good in English this year! Really? Are you curing cancer or helping the homeless? If not, use the adverb well, not the adjective good.
Rules 7, 5, and 2 are strictly orthographic. Spelling is not normally considered grammar. It's also not tested on the SAT. And of these, the capitalization rule (#5) is incoherent. The most important subject? Huh? Is this a muddled attempt to define proper nouns?
Rule 6 is also orthography, although it is punctuation in the service of syntax, so I'll concede this one.
Rules 4 and 3 both concern pronoun case. Rule 4 certainly is an issue on the SAT, but it's cast very narrowly: object case after prepositions. There's no hint that it's part of a bigger issue involving case within conjoined phrases.
Rule 3 is one of those prescriptive bugaboos, but it's a very minor point. Even if you believe that people should say This is she, predicate nominative constructions are very limited in scope in English. Apart from This is she, It is I, and the like, the only other way you can get predicate nominative pronouns is with a cleft construction that can usually be better expressed otherwise.
So you can write something like
"It must be he who compiled this asinine list."
But under most circumstances, you're probably better off writing
"He must have compiled this asinine list."
Why is something this trivial number 3 on the list? It's a waste of students' time. It certainly seems to be irrelevant for the SAT. I have looked at the writing sections every released SAT, and six PSATs to boot, very closely, and I have not run across a single instance of a predicate nominative. Nor would I expect the test-makers to waste their time on such minutiae.
Rule #1 is a specific usage point (good cannot function as an adverb in standard English), unconnected to any larger grammatical principle.
That's the most important grammatical principle this guy has? What about subject-verb agreement? What about pronoun reference? What about parallelism?
Sure, we can argue about the relative importance of different items, but grammar is not a disconnected list of some teacher's pet peeves. It's an elegant, intricate system for creating meaning. Teaching it the old way is not a return to some golden age. It's a recipe for breeding another generation of students who look at grammar with fear and confusion.