So for a concrete case to consider, the first statistics Rosling presents are about relative child mortality rate in 5 selected pairs of countries. The second are the result of a small informal test of his own students' knowledge of the first.
I don't think it takes an anthropologist to analyse whether Rosling's numbers are verifiable or not. He certainly does not provide evidence in the TED talk that they are correct, in that sense they clearly are not verifiable from the talk. If that's what your friend means then fair enough. A TED talk is not an academic publication.
I'm not aware that the field of anthropology rejects in principle the notion of child mortality being a real thing. So Rosling's comparisons could be checked against other sources. It would not be appropriate to cite Rosling in a paper if you needed a source for the relative child mortality rates of those pairs of countries. You would go to WHO and/or national medical reporting, and pay close attention to their methodologies. In that sense they can be checked, and either verified or falsified. Will your friend give you long odds that Rosling has those pairs of countries in the right order according to WHO or his preferred source? Thought not ;-)
Rosling also hasn't really demonstrated anything about the state of knowledge of Swedish students, although the insinuation is that he has. He (serious-jokingly) says that there's a role for him to teach them something. His small trial is sufficient to support his small claim. His methodology is simple and obvious enough that the test is repeatable with other groups. There's no problem of fundamental science here. I don't know (and I don't think it matters) whether he's ever published on that particular result, but it's probably never appropriate to cite because it's such a limited and specific claim of no general interest. And might be cherry-picked.
Next he shows off his visualization software. The important thing to note here is that this is not an attempt to publish academic conclusions on fertility or life expectancy. He's using UN data (about which a great deal has been written elsewhere), to motivate the use of a particular tool, to combat what he believes are out-of-date general intuitions about that data. His hurried narration makes no scientific claims beyond the fact that a large group of what would be called "developing" countries used to have high fertility and low life expectancy, and now don't. And that AIDS reduced life expectancy in Africa. Neither of those is really controversial enough to really warrant further justification in this context: like the comparisons between pairs of countries it just provides something for you to go away and look up if you want serious corroboration.
There is nothing really to verify or falsify beyond his claim that anyone holds this intuition about developing countries in the first place. That is substantial, it's presented as novel and, even worse than his informal student study, he doesn't quantify it, let alone support it. He's not properly publishing a scientific conclusion, but then I'm not persuaded that he's pretending to.
And so on. In the next comparison US vs. Vietnam, I think he says "by the end of the year" when he means "by the end of the decade". Slip of the tongue, should never be allowed to stand in a carefully produced, copy-edited, reviewed scientific publication, but there it is. So in a sense, no, public speaking is not credible at all since such errors are far more common. In another sense, does this mean there's something wrong with public speaking?
One cannot cite (or even trust) the content of a TED talk as if it were the content of a peer-reviewed journal. TED doesn't do that. Neither is there AFAIK any fact-checking other than what the speaker does or arranges. You can treat the content of TED talks the same way as you'd treat the content of any public address by that speaker. So if you were writing a paper that for some reason needed to know what Rosling specifically says in public, then using his TED talk as a source might be reasonable. Otherwise, not so much, but then why would you want to?
the talk is no longer a scientific talk, and the real scientists are clever enough to stay away from these presentations
I think that's akin to saying that TV documentaries are not scientific talks and real scientists are clever enough to stay away from them. It's true that TV documentaries are not journal articles. It's true that there can be fakery and stupidity in their vicinity. However I don't think it's true that no real, clever scientist can get involved. They must distinguish the activities of "publishing research" and "popular education", and avoid claiming that one is the other, or applying the standards of one to the other.
TED talks are as credible as the individual speaker. The fact that TED has "chosen" them should lend no authority at all but probably, unfortunately, does. As for the credibility of TED as a forum, I don't recall what wit here in the UK observed that the country has very many people who would turn down an honour in principle, but tragically are never offered one. I suspect the same may be true of TED -- there are various reasons you might not want to do it, among them that the content of the talks tends to be over-trusted by people who enjoy TED and its speakers. These reasons are not universally applicable, and in any case apply most strongly to those who won't be invited.
Aside from giving talks, experts in the field might choose to watch or not watch the presentation on the same basis they choose any popular presentation of their field. On the one hand they might be interested to know what outsiders are hearing. They might enjoy the speaker. On the other hand, what are they going to learn? Everything shown will be either commonplace in their field or else the specific and perhaps controversial views of the speaker, that an expert could better assess by reading their publications than by seeing the simplified popular version.