Is the phrase "very delighted" ever "wrong"?

You can certainly come up with theoretical arguments against it, but "very delighted" is delightfully common. Ngram Viewer says it's more common than "very much delighted" and about half as common as "absolutely delighted."

More importantly, it's quite frequently used by native speakers. Some examples:

• The late Queen Elizabeth
• Barack Obama
• Ronald Reagan
• Justin Trudeau
• Martin Luther King

Edit: according to Ngram Viewer, "absolutely delighted" is far more common than "very delighted" in British English, so its acceptability may vary between dialects. That said, Queen Elizabeth used "very delighted" at least once (see above), as has Prince William.

For those calling it a poor stylistic choice, the MLK example above is from the opening of "Beyond Vietnam," a masterpiece of oratory and one of his greatest speeches.

Adjectives like amazing, awful and boiling are also non-gradable. They already contain the idea of 'very' in their definitions. If we want to make extreme adjectives stronger, we have to use absolutely or really:

The argument that "delighted" doesn't take "very" is based on "delighted" being an ungradable rather than a gradable adjective. It carries implications that are closer to words like "dead" and "pregnant". People can't be a little bit dead - they either are, or they're not.

As such, the intensifier would be a pure intensifier and not a qualifier or quantifier.

Definitions seem to be looser in current use - it wouldn't sound particularly strange to ask "How delighted are you?", in which case "Very delighted" would be fine and it might be considered pedantic to take issue with it.

The grammarians that you quote are on weak ground. There are other words that contain the notion of intensification. To give two examples, there are:

Cambridge
Intense
Extreme and forceful or (of a feeling) very strong:

Macmillan
Brilliant
Very intelligent
very skilful, impressive, or successful

Both these words, and others, are often intensified by very.

The grammarian argument seems to be that a word deriving from intensification (very, extremely, gravely, even absolutely) of another word must not itself be intensified. This restricts the scale of intensification arbitrarily. It also expects users of a word to consider its antecedents, meaning and etymology before intensifying it.

Degrees of intensification are acceptable;
It is appalling (=very bad) that grammarians should adopt such an attitude and,
(furthermore)
it is very appalling that they presume to proscribe our usage.

As a matter of usage rather than semantics, they are clearly wrong, very wrong, very very wrong, extremely wrong, and absolutely wrong.

According to Wikipedia:

The Longman Dictionary of Contemporary English (LDOCE), first published by Longman in 1978, is an advanced learner's dictionary, providing definitions using a restricted vocabulary, helping non-native English speakers understand meanings easily. It is available in four configurations

As is the case with learner's dictionaries in general, LDOCE has usage guidelines that have to be prescriptive. So what they mean by "don't say" is not always that the construction is not grammatical per se, but sometimes that it's not idiomatic or natural among native speakers.

This Ngram shows that "very delighted" is the least productive among "[adverb] + delighted". (Although it is shown above "not delighted", "very delighted" is actually less productive than even "not delighted".)

Just because famous people have uttered this expression doesn't mean it's necessarily idiomatic or natural. Native speakers, famous or not, do utter unidiomatic or unnatural things from time to time. What's clear is that the expression will never make it to the article of The New York Times or some such, unless in a quote.

I don't agree with the Merriam-Webster blog post you've cited, and it is not even relevant here, because "very pleased", unlike "very delighted", is the most productive of all "[adverb] + pleased", according to this Ngram. Note also that "much ecstatic" is even less productive than "very ecstatic", according to this Ngram, which perhaps tells us that adjectives converted from past participles are more likely to be collocated with "much".

I think you may have guessed it right when you said:

The definition of delighted is "very pleased and happy." Perhaps very is already implied, so another intensifying very would be redundant?

Note "very ecstatic" is similarly unproductive, according to this Ngram.

Finally:

But then why does absolutely work?

I guess "absolutely" as well as other emphasizing adverbs such as "so" and "quite" emphasizes the following adjective in a different way than does "very".