Their argument about contractions, which they do not relate to 'between you and I' is interesting.
Yes, if by 'interesting' you mean 'piffle'...
It would be ludicrous to attempt to justify the correctness of 'between you and I' on the grounds of an analogy with contractions,
I'm relieved to hear that we can at least agree on that much...
but H& P make no such attempt
Oh, they don't, do they not?
Well, pausing just momentarily to wonder exactly what constitutes 'making an attempt' in your view, I would respond thus: on p10, lines 1-5 "The sequence ...[3a] and [3b]" effectively assert that there is a precedent for reckoning that rules of morphology regarding a given pronoun as a single form can differ from those regarding the same pronoun occurring in a compound, and that that precedent is the so-called rule of grammar cited at the bottom of p9.concerning contractions.
In short, they stop just one hair's breadth short of saying "[3b] is grammatically perfectly OK in our view, and you pedantic old fusspots who've been objecting to it for years had better jolly well knuckle under and accept it!"
Yes, I think that definitely rates in most people's book as an 'attempt'...
And now for some reasons why I find their argument ludicrous:
1) If you're going to support a case for suggesting/implying/hinting at (call it what you will) the grammatical acceptability of one construction by analogy with another, you had better ensure that your analogy actually concerns grammar. The use/non-use of contractions is an issue of either pronunciation or orthography, depending on whether we are dealing with the spoken or written form.
Whether one elects to carefully enunciate the 'you are' of their [4b] as two distinct words or to lazily conflate them into something sounding like (and consequently recorded in written form as) one makes absolutely no difference from the point of view of grammar: 'you' and 'are' remain, irrespective of such incidentals of execution, two grammatically distinct words, namely one pronoun and one verb respectively.**
2) If you're going to support a case for suggesting/implying/hinting at the grammatical acceptability of one construction by analogy with another, you had also better ensure that you compare like with like. In suggesting a parallelism between
[3a] They invited me to lunch.
and
[4a] I don't know if you're eligible
they are implying equality of status between [3a], a formally flawless sentence (definable, for the purposes of this analysis, as one acceptable to all users of the language at up to and including the highest level of formality) and a flawed one, [4a], since the use of contractions will not be considered acceptable by all users in all conceivable circumstances (e.g. in legal documents).
Thus their implicit assertion that the sequence [4a] to [4b] - in reality that of slightly flawed to even more flawed sentence - can be held up as in any way mirroring that realized by [3a] to [3b] (flawless to flawed) is simply untenable.
3) (And I've saved the most ludicrous bit till last) If you're going to support a case for suggesting/implying/hinting at the grammatical acceptability of one construction by analogy with another, you had better not shoot yourself in the foot by rejecting analogy in principle as a valid basis of grammatical argumentation (see p9, paragraph beginning "Prescriptivists, however,...").
So, overall, I would say that the word 'ludicrous' sums up their position rather nicely!!
**I would also be grateful if Messrs. Pullum et al. would explain at precisely what point between a perfectly enunciated "you are" and a completely contracted "you're" - since a little experimentation shows there to be any number of phonologically indistinct midway stages between the two - sentence [3b] goes from being correct to incorrect.
