The OED in its entry for "time" has at P5a:
(b) Chiefly poetic and literary. In the fuller form many a time and oft (also often) (and variants).
c1300 (▸?c1225) King Horn (Cambr.) (1901) 1070 (MED) Horn bad vndo softe Mani tyme & ofte; Ne miȝte he awynne Þat he come þerinne.
1446 in L. Morsbach Mittelengl. Originalurkunden (1923) 34 (MED) Y herd my seyd mastyr sey in hys good lyve..mony tymes and ofte, that he neuere seled dede..to the seyd john Rope.
1996 ‘J. le Carré’ Tailor of Panama (1997) iii. 39 No suit should be worn two days running... As I'm sure your good father will have told you many a time and oft.
It is worth noting the singular in the earliest quote.
P5e. times without number: on countless occasions; very frequently.
1658 T. Pierce ᾽Εαυτοντιμωρούμενος: Self-revenger To Rdr. sig. *** Times without number he calls me proud and insolent.
2000 Econ. & Polit. Weekly 16 Dec. 4457/1 This farce has been played out times without number at the national level and in individual states.
And then there is the Google Ngram for Time and again,times and again,time without number,times without number
In which you will note that “time and again” is more frequent than “times and again” but that “times without number" is more frequent than “time without number”.
So, how are we to understand “I told her time and again not to touch it”?
“I told her again not to touch it” is clear enough – “again” is an adverb.
“I told her time not to touch it”... This is incomprehensible if “time” is a noun… but not if it is considered an adverb = on occasions or on an occasion.
Then we have
P2. With a following adverb.
|b. Originally U.S. time and again (also time and time again, (now less commonly) times and again): repeatedly; on many occasions; very often.
It is obviously an adverb.
1821 Jrnl. Deb. & Proc. Convent. to revise Constit. Mass. 48/2 Application was made, time and again, relative to the College.
2009 C. Ackerley in W. Van Mierlo Textual Scholarship & Material Bk. 109 Time and again she had to make difficult decisions about disputed words and phrasing.
Here “on occasion and occasion again” fits well.
And then we have:
c. time(s) and oft (also often): = many a time and oft at Phrases 5a(b). Now rare.
1791 W. Taylor tr. G. E. Lessing Nathan the Wise (1805) iii. 130 And have not I too said so, times and oft.
1798 Musical Banquet 122 Time and oft, dress'd lamb fashion, I zeed an old ewe.
1972 P. M. Fraser Ptolemaic Alexandria I. x. 658 A floating islet..which sailors from the Cyclades and the Saronic Gulf saw time and oft in the waters of the northern Aegean.
in which we again see an earlier plural use.
And here we have the answer:
Despite Google Ngrams, time can be understood either as an adverb "on an occasion" or as many a time and “many a time” = many times and is adverbial.
You also asked
How are we to understand the following sentence as being grammatical? If it isn't, why should we excuse it's not being so? [...] If indeed incorrect, how are we to tolerate ungrammaticality in a work of prose?
We do this by realising that the person has not been born who speaks and writes any language 100% correctly, 100% of the time. We realise that we all make mistakes.
Of course, the pedant in us all may rail against such egregious faults that are clearly a sign of a lower intellect than ours, but then we must congratulate ourselves on seeing the error and thus satisfied, eagerly anticipate finding the next.