The original manual was extremely vague about how the Protoss' "primal link" worked other than to suggest it was a highly efficient application of telepathy (as opposed to something wholly different from telepathy). The Khala referred to the science and religion built around it.
As the retcons piled up, this became increasingly blurry. The link and the Khala became synonymous in the writing, and this led to a number of what I consider plot holes.
According to SC2, the tal'darim don't have the Khala/link because they joined it when Khas founded it. The problem here is that the Protoss always had the link and the tal'darim supposedly left before the Aeon started.
For the sake of simplicity, I will refer to the link in all historical eras as the Khala and the religion as the Path of Ascension.
Were the tal'darim part of the first Khala and then breakaway from it? In that case, their backstory claiming they never joined is wrong: they broke away from it just like the nerazim did.
When the tribes broke the first Khala, did they just break the inter-tribal Khala or did they break the intra-tribal khala too? I ask this because, prior to LotV claiming the khala was evil or whatever nonsense it was saying, the lore made it clear that the khala was the only thing preventing the protoss from fighting endless war (not unlike the overly emotional eldar in warhammer 40k, which they were clearly modeled after). So if the tribes broke their tribal khala, then they would attack their own families eventually, which would have prevented them from surviving long enough to fight an interstellar war with other tribes.
The protoss evolved with the khala, so I have to assume the rest of their psychology is adapted for it and doesn't work correctly without it. This is comparable to how humans suffer from mental illness if their brains are not working correctly. Presumably the nerazim have developed a means of maintaining their sanity and emotional control, since the protoss developed new psionic sciences and such that allowed them to break the khala in the first place.
Anyway, the only consistent explanation I can think of is that the tribes always had some form of khala. The first empire formed a super-khala that connected the tribal khalas, which was shattered during the Aeon. The tal'darim and nerazim are oddities, since they reject even the tribal khala.
I can understand the nerazim rejecting the link because of their disdain for the conclave and such (and practical reasons like studying the new field of void psionics), but the tal'darim are much more difficult to explain. They use sundrop to prevent development of the khala, or at least that was the explanation for one tribe of them (for simplicity we can assume it is a widespread practice to explain their intact nerve cords). If they worship the xel'naga, surely they would embrace the khala? I would think that somewhere along the line their religion got the wrong idea and accepted values that their former gods would have rejected (which is actually quite common in real religions, since they don't stay static).
Feel free to share your own thoughts, observations, explanations, etc. This discussion will probably be contradicting the canon, given that the canon isn't really reliable or consistent in this situation.