What surprises me is that he's not ousted by the board yet. Surely at this point he's causing more trouble than he's worth?

Part of spez's job is to be the lightning rod for controversial decisions. The board I'm sure is pushing for the same things (increases in pricing, driving users to the official app) in order to boost metrics before the IPO. Aside from the somewhat pointless AMA where his frustrations came out a bit too much, if you assume that the effective removal of API access had to happen, what do you think he's done wrong during this?

Many take exception to his handling of Reddit’s relationship with Apollo and Christian Selig specifically.

Steve Huffman has reportedly told employees that Selig threatened Reddit. Selig posted a (perfectly legally recorded and disclosed) call recording showing the alleged “threat” was a misunderstanding over which the Reddit employee on the call apologized immediately.

Huffman serves in an official capacity at the Anti-Defamation League. People are (rightly, I think) critical of his handling partly in light of that.

Maybe he knows where all the skeletons are buried, so the board removing him would be to difficult (or they're not ready to pay his golden parachute/buy his silence)