Wonder if any of those counseling the OP to "just ride the tire until . . . " have dropped hundreds of bucks for a set of wheels that were slightly lighter than their existing, perfectly good wheelset. Priorities!

I admit to being a cheapskate about any number of things (e.g., I still occasionally wear a nylon Rampar-branded cycling jersey that I bought when I worked at a Raleigh dealership in 1977), but not about damaged tires.

I also get wanting to save a tire with good tread but this tire is DEFINITELY no bueno. I'm a notorious bad tire denier but even I wouldn't ride that.

To the OP, swap out the tire (check the other tire too), make sure the hub is in the dropout properly and squarely, check brake pad alignment one more time and you'll be good to go.

Iride01 I recognize those tires, they're Panaracer Pasela's, one of the better 27 inch tire options still available. They're no top shelf tire but still of solid quality, Panaracer makes a good product.
Not to be rude but I would argue that there is a wrong thing to do and that would be to continue riding the tire. Too much of that sidewall is fraying for me to feel comfortable recommending to anybody that it's safe to continue using it. Of course, we're all welcome to our opinions and the OP will do whatever they deem best.

Co-ops and nonprofits will often cannibalize one bike to fix another. I suspect that is what happened here. And typically by volunteers. Doesn't matter, you need new tires. I am guessing that you paid very little for this bike.

+1 Didn't read the rest of this thread but that was my first thought. **** happens. My city bikes use Paselas. They are very prone to sidewall damage if anything touches them. Brush a curb on a careless stop at a crossing. Oops. I now have the OP's tire. Brakes haven't been looked at for too long, the pads wore and rim contact moved up just a tiny bit too far. Oops. Rear hub nut/quick-release not tight enough? Tire touches chainstay? Oops/

There are tires that do far better for sidewall abuse. Those delicate sidewalls (for unintended contact; they roll forever if you let them) are what give those great city tires their wonderful characteristics. I rode those same tires, just a larger size down the Trask River Trail to the Oregon Coast over the coast range; going down steep 1000' descents on 1-1/2" to 2" "gravel" and they simply didn't care. That was 2017. I don't ride those wheels much and the tires are still going strong.

I had these on my road bike for years. They were great tires. My only complaint was that after a long tour in the rain, the sidewalls were permanently stained by aluminum run-off from the rims (rim brakes). Other than that, no issues.

A couple of my bikes came with those LAME flimsy sidewall tires around 2002. They totally cracked by the tread after only 3 years. Do NOT ride one inch.
I fail to see the attraction to the UGLY things anyway. Otherwise the tread would be rideable. Replace with solid black sidewalls, Schwalbe makes the best lasting rubber by FAR.
OTOH My co-op bought in2017 CCM had 30 or 40 yo nylon tires that were pretty hard. I rode the front one 1930 miles with new tubes.
The bead was crumbling when I took it off.

That damage looks like dry rot to me - put away soaking wet, parked long term underinflated.