tl;dr: Players did indeed die from having their brains fried. Otherwise, it would be odd for Kirito to never be told and to make statements suggesting that he doesn't know.
A general meta-observation
Firstly, I should note that it would be very odd story-telling if we (and Kirito) were deceived about players having their brains fried and never were told that this was actually a lie, given how the "real death" factor was so intrinsic to the set-up of SAO. It's normal for stories to have plotholes or to leave some things unexplained; this, however, is more an issue of some sort of Occam's Razor for stories.
Observations from SAO
Halfway through the first episode, Akihiko states that there is no logout option, and that removing the NerveGear will result in death. He then pulls up a bunch of news articles and footage about the deaths that have already occurred. However, perhaps Akihiko is just bluffing and making fake news articles.
If Akihiko were lying, then it wouldn't make sense for the players to be trapped so long, as suggested by David Mulder; people outside could just remove players' headsets. Perhaps people outside wanted to play safe after the announcement (despite the reduced credibility, since the 213 who supposedly died would still be alive) or heard a different version (e.g. "After this announcement, everyone who tries to exit the game environment will be killed."). However, this can only be the case if they had cause to believe that NerveGear was capable of killing people.
Moreover, as suggested by Killua, these scenarios would seem inconsistent with Akihiko's conversation with Kirito after SAO is cleared:
Akihiko: Just a moment ago, the remaining 6,147 players were logged out.
Kirito: What about the ones who died? What about the 4,000 who died?
Akihiko: Their minds will never return. In every world, once you die, you're gone.
By this point in time, Akihiko sounds relatively forthright, and it would be really strange for him to omit information at this point if Kirito is going to find out the truth once he logs out anyway. However, it's still possible these people instead all died "from malnutrition or the extreme stress of thinking it's the real deal". (This still leaves the 213 who presumably would not have actually died at the beginning. Again, it would be very odd story-telling if they didn't actually die and we never knew.)
It's thus worthwhile to look at Kirito's interactions after he leaves SAO.
Post-SAO, in-universe evidence
In the first episode of the second season (about 13 minutes in), Kirito questions the government official who approaches him if he was sure that the players of Gun Gale Online have really been dying of heart attacks after being shot in virtual reality-like circumstances. He wonders if the GGO cases are similar to what happened in SAO:
Kirito: You're sure it was heart failure, right?
Kikuoka: What do you mean?
Kirito: [flashback to an image of the NerveGear headset] There wasn't any brain damage?
This suggests that players could indeed die from brain damage after being killed (or having had their headsets removed) in SAO. (Kikuoka even later mentions NerveGear's ability to destroy one's brain via its microwaves.)
Moreover, it probably was the case that players did die from this mechanism. If not, it would be odd for Kirito to not be aware of the fact, or for Kikuoka to not correct him (e.g. "Sure, NerveGear is physically capable of destroying brains, but that's not what happened in SAO."), when this scene takes place more than a year after SAO was cleared. (SAO is cleared in November; the GGO arc starts in December of the next year.)
After all, Kikuoka would presumably be aware of such cases as a government official involved with the SAO case. Similarly, Kirito was involved in the investigation---by the GGO arc he has told Kikuoka everything he knew about the SAO incident---so it would be expected that he would know of such specifics a year after the case.