Word to describe using sarcasm or a similar approach to lead someone to recognise their own mistake

I would use "facetious" in this context. I define being facetious as "just messing with (something or someone)." It's not quite sarcasm, but it's a close relative, IMHO.

"facetious" is the best reply thus far, but, with all due respect, you were being a smartass...I don't know it is allowable in the context, though.

"smart aleck" is the more socially-acceptable flavor.

you were being disingenuous

Pretending to be unaware or unsophisticated; faux-naïf.

or you were being faux-naïf

Marked by a false show of innocent simplicity

You were almost playing the devil's advocate.

play/be devil’s advocate
to pretend to disagree with someone in order to start an argument or interesting discussion
Synonyms or related words for this sense of devil’s advocate
To pretend to do or feel something: pretend, fake, simulate, assume, make believe, put on, affect, play along, keep up, feign... more

I said almost because you did not really say anything that you didn't agree with. But you did intentionally read the sentence incorrectly just to make your friend realize that there was something missing.