why can't you kill with your king if it's in check

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A king can capture a checking piece if the checking piece is undefended (and on an adjacent square).

You can't sacrifice your king.

You can.

For example

Not since 1996.

5.2 b The game is drawn when a position has arisen in which neither player can checkmate the opponent’s king with any series of legal moves. The game is said to end in a ‘dead position’. This immediately ends the game, provided that the move producing the position was in accordance with Article 3 and Articles 4.2 – 4.7.

Yes, it's a draw but the king can still capture the queen.  Are we missing something?

Yes.

Black would have only one legal move, capturing the queen.

BUT that move produces a draw.

Therefore, the game has already ended and Black should claim a draw under article 5.2b without actually making the move of capturing the queen.

Compare if Queen were at Qg7+ instead. In that case, the game would NOT be drawn, because Black would have a choice of two legal moves: capturing Kxg7, which ends the game with draw, or moving out of check with Ke8, which allows the game to go on because checkmate is still possible.

So: in case of Qf7+, because Black has no legal losing move, he has no legal moves at all, under 5.2b.

Haha, ok, that's a funny case.

It's true that after 1.Qf7+ there is no legal series of moves that will end the game in checkmate. So after 1.Qf7+ in that position the game ends in a draw without black needing to capture the queen.

Here is a different example then:

Also note that a King can get out of check by capturing an enemy piece which is not giving check.

What I’d be interested in seeing as a rule change or variant is a king being able to kill one of its own pieces (say a pawn) to escape a mate.