I think "cancel culture" definitely has some left-wing connotations, but I think that's because it came into existence at a time when the left was doing the majority of the canceling. In the aughts, "canceling" was mostly a conservative thing (usually criticizing the war effort) and it was much less egregious (if only because social media barely existed at the time). One prominent, egregious example of conservative cancellation from the time was the Dixie Chicks.

So yeah, "cancellation" has some left-wing connotations, but it's not something that only the left can do. Further, there absolutely are hypocrites who criticize "cancel culture", but who happily try to cancel people they don't like. Hypocrites are bad, but there are still lots and lots of principled critics of cancel culture (indeed, I'm pretty sure most of the Harper's letter signatories are left-of-center). That there are hypocrites doesn't validate cancel culture.

Also, if you're upset about the left-wing connotations, the proper response is not to try to legitimize cancel culture, but rather to persuade your political associates to behave better for sake of the brand.

TL;DR: Canceling is bad no matter who does it; recently it's mostly been the woke left who have been doing it so it does have some political connotations; sometimes people are disingenuous in criticizing "cancel culture".