When you sign an open letter that will be run very publically, the implication is generally that you endorse its content.
It is. And the argument would then be that Chomsky never actually read or understood the context of the letter he signed. Which for someone like Chomsky, seems unlikely.
But falling for a cheap rhetorical trap is? So now we flip a coin on whether he misunderstood the letter or the email.
Possibly. Or maybe he misunderstood neither. And that's the thing. The letter refers to a lecture Chomsky did in which he stated that the right resort to 'hysterics' to delegitimise specific groups. He then goes on to ask whether he still believes they are doing it now. To which Chomsky replied, 'it is happening from Trump down'.
The problem with the argument you're making, or at least the way I perceive it, is it seems like you're saying there is no room for nuance. That if you believe the right use arguments about political correctness to delegitimise their opponents, then that must mean he automatically believes everything the left does is perfectly legitimate. That everything is a zero sum game.
There's something else I know. I'm not sure if you're aware that there was another letter circulating around the same time as the Harper's letter. It was a letter attempting to get Steven Pinker 'cancelled'. Chomsky stated he did not sign this letter and explained his reasons why.
Someone asked him about why he didn't sign this letter. He replied with this:

You notice this response is very nuanced.
Here's the thing: The person who wrote to Chomsky about the Harper's letter could have asked him directly what the reason was for him signing the letter instead of doing it in such a roundabout way. I'm sure Chomsky would have given him a reason. He instead wrote about a lecture Chomsky done and basically asked him if he still agreed with that now. And Chomsky seemed to agree. Of course it was a safe bet Chomsky would agree when framed in that way.