Shameless Mr Inarkiev...

Correct. Carlsens King move was to a legal square and was indeed a legal move. Carlsen had not made a claim that Inarkiev made an illegal move. His move was legal and the game could have continued with Inarkiev being in check. However, Inarkiev stopped the clock, claiming Calsen's move was illegal. The rules could be better written, clarified, as the Chief Arbiter said. The 1st Arbiter was over ruled who declared a win for Inarkiev.
The Chief Arbiter made a fair and reasonable decision to resume the game. If anything, Inarkiev could have been penalized time for making a bad claim.

Interesting thing is that Carlsen won a game that was at one time ruled a loss, a draw and a win for him.

Make moving into check and capturing the king legal. Problem solved. You move into check, you lose. I have played that way OTB it is far better than fumbling around with the pieces and clock. And it is more pure, more realistic in a battle. I think the rule of can't move into check should be completely eliminated. Making illegal moves and knocking over pieces then hitting the clock should be a loss for the first person who did it. Solves that problem as well.

The issue comes about because the player was in check. He made another move instead of moving out of check or capturing the piece. With no claim of a illegal move being made, the game continues as if nothing had happened.

When Inarkiev stopped the clock to make his claim of an illegal move, he knew exactly what he was doing, as his immediate claim was of an illegal move made by Carlsen. (Inarkiev's King remained in check) and asked to be declared the winner.

The OP is spot on ... Shameless

Allowing the capturing of Kings in blitz games is a good option and avoids a lot of disputes. Unfortunately, it is not in FIDE Rules.
Club games are often played like this.

Well when Carlsen gets duped with this then I hope chess fans will push for rules changes. If the first illegal move noticed wins, then capturing an illegally moved piece should be legal including the king. And a piece knocked over and hit clock should reset opponents clock to add at least
+1 minute, regardless of what piece it is. I do recall a Womens world championship match coming down to time odds where the defending champ had a piece knocked over and clock hit and she lost on time while fixing her opponents mess. I also saw Nakamura have to fix one of Carlsens knocked over pieces with no time added. They both kept playing but Naka lost time.

The rules have already been changed. From 01.01 you wont lose after making one illegal move. After the first offence your opponent gets time added to his clock. You lose on the second offence.

In blitz and rapid the first illegal move claimed wins, in classical the second.

Capturing the king as „evidence“ has been abolished and is considered a illegal move itself. There are not few who don’t parry a check, the opponent „takes“ the King and they apply for a win Ernesto-style.

Just google „chuzpe“, but you can call it simply „stupid & shameless“, because it lacks esprit.

In blitz and rapid the first illegal move claimed wins, in classical the second.

That's the latest change. What is not said: what if a false claim is made in rapid and blitz? If a ruling goes against the claimant, is it declared a loss or is a penalty of time incurred, as in classical?

Such rulings can be left to the discretion of the Arbiter, in "the best interests of fairness" as in this case. The Chief Arbiter decided the game should resume at the point Inarkiev was placed in check.

But wondering if technically, with the new rules, since Inarkievs claim was disallowed, could the Chief Arbiter have declared a loss for Inarkiev? I think he bent over backwards and gave him the benefit of doubt, that he did not realize he was in check and found a reasonable solution (that Inarkiev rejected.)

Right, according to my rules, Carlsen wins. Good luck making sense out of a mess that can be avoided using strict rules. Why show any mercy whatsoever to a player who makes an illegal move first, and then hits the clock?? there should be 0 tolerance. Instant loss of game. If can move my queen through a teleport to take your queen on move 1 for checkmate, then I should lose? Assuming you premoved D5 or smthg.
Not oh no you need to adjust.... You guys need to draw realistic lines. Don't give hustlers and cheaters a means to time swindle the best players who have honor and follow rules.