Are the rules regarding absolute adjectives too absolute?

What you are doing in the examples very unique, more perfect is using an adverb to intensify a strong adjective; it seems a fairly good rule of thumb that this is often stylistically inadvisable. There is a certain inflation of the intensity of the adjectives, and the adverbs can be accused of pleonasm (as in "a triangular triangle"). This rule is often referred to on both sides of the Atlantic, and in many other languages too.

However, your other examples do more or less the opposite: you use adverbs to weaken (or qualify) the adjective in almost exactly, virtually all, and practically infinite. That is quite different: there is no inflation of the intensity of the adjectives themselves, nor are the adverbs pleonastic.

Other concerns may arise, though, in the case of absolutes and classifications (I like Fumblefingers' categories). Sometimes it is not very satisfying to qualify the unqualifiable, as in she's a bit pregnant: she's either pregnant or she isn't, so it is probably better to say she looks a bit pregnant, if that's what you mean. I think that's almost exactly the same thing, while perhaps acceptable, is usually not to be preferred over that's almost the same thing: the adjective feels a bit wordy and is made redundant by the adverb. The examples with all and infinite are what Fumblefingers calls absolutes, which I think should be qualified without qualms, where appropriate.

Not sure exactly what's being asked here. Do English language teachers still tend to warn against using grading adverbs with non-gradable adjectives? Probably, because the reason hasn't changed.

Grading adverbs (e.g. extremely, fairly, rather, very) vary the intensity (grade) of adjectives.

Non-gradable adjectives describe qualities which can't normally be varied because they're extremes (e.g. freezing), absolutes (e.g. dead), or classifications (e.g. nuclear).

But rules are meant to be broken — it would be a sad day for language if we weren't "allowed" to say, for example, "I feel very alive this morning".

TL;DR: Nothing's changed. Learn the basic principles first, then break the "rules" whenever it seems logical/eloquent to do so.

It’s not a rule of grammar, but a matter of lexical choice. The question turns in part on whether the adjective being so modified does indeed represent an absolute value. Unique certainly has its origins in the notion of oneness and there are those who believe any departure from such a notion will lead to the collapse of civilization as we know it. The prophets of doom conveniently ignore, or pehaps have never known, that unique has been susceptible to modification from the early nineteenth century. The same can be said of perfect. In 1623 Bacon wrote:

Those works, which I had formerly published, . . being retractate and made more perfect.

while in in the late eighteenth century Gibbon wrote:

The heir most gratefully subscribed an agreement, which rendered my life-possession more perfect.

I don't seem to see any reference to absolute intensifiers here (apart from in the English Club link), so at the risk of repeating a bit, here's how we teach it in EFL (BrE)

With gradable adjectives (happy, tired, surprised etc) the main intensifiers are:

very, really and quite (meaning partly). Later we'll add extremely, terribly etc

With ungradable adjectives (delighted, exhausted, amazed etc) the main modifiers are:

absolutely, really and quite (meaning 100%). Later we'll add totally, completely, utterly etc.

At advanced level we might divide ungradable adjectives into extreme adjectives and absolute adjectives, and show how these intensifiers strongly collocate with certain adjectives:

You're quite right; that's utterly wrong; it's perfectly ridiculous; it's absolutely perfect etc

In fact these are not really modifying the adjectives so much as emphasising them. So what about unique? Well, I'd never say "very unique". Not because of 'the unique squad', but for the same reason I wouldn't say "It's very fabulous", it doesn't sound natural English to me; we don't normally use very with ungradable adjectives. But I might say "totally or absolutely unique". The unique squad probably still wouldn't like it, but at least it would sound natural English.

However, I've just checked in MWDEU, and to my surprise they allow very unique, no less a writer than George Elliot having apparently used it - "A very unique child, thought I". It still sounds unnatural to me, but maybe I'll have to rethink.