sidewall tire cut

Yes. Very. Sidewall flex is a major issue here. With a smaller-than-original thickness of rubber involved, the issue is magnified. You've got a highway-speed blowout (as in "catastrophic failure") waiting to happen.

Yes, it should be replaced. As "Immediately" as possible. There's a reason that a tire place won't "fix" such damage - Namely, there is no truly effective fix. There are several "fixes", but they're all about as effective as a band-aid over a slit throat - "it looks OK, but it isn't dealing with the real issue."

You could have a fast and complete loss of pressure from a cut like that. Replace the tire asap and don't drive at highway speeds 'till you do.