Nobody can know, of course.
That said, I'm afraid I'm not convinced.
So far as I can see, you need three premises:
- there was linguistic diversity in Iberia
- that linguistic diversity preceded the kurgan invasions
- in terms of diversity, Iberia was representative of Europe as a whole
All three premises, in my opinion, are highly questionable.
Regarding the three families and diversity in Iberia:
- it's not even clear that Basque was present in Iberia in prehistoric times; our earliest and most voluminous attestations are from France, not Spain, and it appears that in at least part of its later Iberian territory it was overlaid on a Celtic (or similar) substrate. While it's possible that it was present all along, we can't really assume that for sure, as I understand it.
- Iberian appears to be related to Basque. Sure, we don't know that, because we don't know much at all about Iberian. But what little we do know all seems to point to correspondences with Basque in names, in morphological elements, and in numerals. Now, it could be that there was massive borrowing from one direction or the other, as well as sprachbund effects. We can't disprove that. But the more parsimonious hypothesis would surely be that all the resemblances are due to the languages being related, and not even all that distantly.
- Tartessian does look different. But again, we know so little! Imagine trying to prove the existence of Indo-European, when your only resources are a grammar of modern Bengali and some fragmentary inscriptions in Old Irish! Bear in mind also that related languages can undergo substantial surface differentiation, particularly in a prehistoric context where travel may have been less widespread. (look how difficult it's proven to demonstrate large families in Amerindian).
So it's not clear to me that there WAS such diversity in Iberia.
Regarding the second point: we don't know that ANY of those languages are Neolithic in origin, let alone Mesolithic. What we do know is that by the time of our earliest attestations the region had already been invaded from at least two directions - Celts had already crossed the Pyrenees, and Greeks and Phoenicians had already arrived by sea. There is no particular reason to think that neither the Iberians (who seem related to the Aquitanians living north of the Pyrenees) nor the Tartessians (who seem anomalously developed for the area and had extensive contact with the Phoenicians and Greeks) migrated to the area not long before historic times. NON-IE doesn't necessarily mean PRE-IE.
In particular, while we could see the Basques as relicts, we could also see them as invaders. Our first depiction of them (thanks, Caesar) is not as peaceful isolated mountain dwellers in Iberia, but as a warlike race of horsemen in France. Genetically, although there is an unusually high degree of mesolithic and neolithic ancestry in the basques, they don't seem to stand to one side of the rest of europe, the way the Sardinians do, and apparently there is considerable genetic continuity across the atlantic region. We (seemingly) know that, for instance, the British Isles were invaded by the kurgans long before it is feasible to imagine the celts, per se, being there. That atlantic invasion could then be linked to, for instance, the prevalence of R1b and the Bell Beaker culture. In short, I think it's actually more attractive to see the Ur-Basques (Ur-Bell-Beakers in this hypothesis) as allies of the PIE - a precursor invasion that was then overrun by the second wave.
Even if we did see both Basque-Iberian and Tartessian as "native", in the short term, we know that Europe suffered at least two massive invasions from the middle-east with the Neolithic transition - the LBK by land, and the Cardials by sea. If these were indeed two language families (though genetically they're closely related, so that's far from certain), it wouldn't be too surprising to see two language families in Iberia, a region that could feasibly be reached by both routes (Tartessians as Cardial seafarers, Iberians as LBKs come over the Pyrenees).
Finally, regarding the idea of Iberia as representative: it seems the exact opposite to me.
For one thing, mountainous peninsulars in general aren't somewhere to look for representativeness. Even if it were true that there were 3 neolithic stocks in each peninsular, there might still only be 10-15 stocks in all of Europe, rather than 25-30. Consider how many apparently distinct families there are in California, compared to those of the great Plains... (and Iberia was much,much less diverse than California, at least by historic times).
But there are also specific reasons to doubt the representativeness of Iberia.
First, if we're talking about the Mesolithic situation, there is a big difference between the peninsular refugia and the thawing tundra. The tundra seems to have been rapidly repopulated by a single culture, but that doesn't necessarily mean that the same people took over Iberia: it would be more likely that as those people spread out from the refugium, they left whatever diversity was there to its own devices - repopulating a thawing wasteland with seminomadism is much easier than overwhelming an established fish-eating population hotspot. [not that I imagine there were many stocks surviving in that refugium anyway, but it's possible]. The Tartessians in particular are as far away from the Mesolithic expansions as possible, so they could well be only very, very distantly related to most of mesolithic europe.
Then, regarding the Neolthic invasions, Iberia is at the exact opposite end of the continent from the source of those invasions. If anywhere retained pre-neolithic languages, it would be Iberia. Likewise, Iberia would be one of the most likely places to see settlement from both directions - or indeed from a third, precursor group (settlers across the Med before the main Cardial expansion). So Iberia has particular reasons to show much more diversity than the rest of Europe - and frankly I think the interesting thing is that it doesn't show MORE diversity!
So while the crazy-quilt hypothesis is obviously not something we can currently falsify, and almost certainly never will be, it seems to me that it requires a whole series of unparsimonious assumptions, and is less likely true than untrue.
But the river tripped on her by and by, lapping
as though her heart was brook: Why, why, why! Weh, O weh
I'se so silly to be flowing but I no canna stay!