Fixing or Buying a new iMac?

+1 replace it. What is likely wrong is the GPU which is integrated onto the main logic board and is not replaceable. It is also quite obsolete at 11 + years so my 02 cents is to replace it. I would strongly consider a new Mac mini with a 27" display, you will get far more power than a 2013 iMac and you can select the 27" display you want. For about $1500-$2000 you can have a very nice solution.

Replace. 11 years is a good long life and there is too much uncertainty on GPU vs Display failure for what even needs fixing.

P.S.: Easy display test: does the iMac work with an external monitor?

You can use a mini display port to HDMI cable and a TV if you don’t have a traditional external monitor.

Vertical lines usually indicate a problem with the LCD Panel. Since you can remote into the system without issue seems to confirm it. You could try connecting an external display to the iMac as further confirmation. The faded & rounded corners also point to an LCD panel issue.

Apple refurbs are an even better option. They come with a full one-year warranty and qualify for AppleCare+ extended coverage.

Refurbished Mac - iMac - Apple

Over half our family's considerable inventory of Apple products are Apple-direct refurbs. All have been superb performers. Zero cosmetic issues either.

The Apple online store is the ONLY reliable source of true Apple refurbs. Any others advertised as "factory refurbished" should be taken with a ship-load of salt! One person bought an iMac represented to be factory refurbished from an auction site. He posted here wondering why Apple would use duct tape to secure the display!

OWC is usually forthcoming in saying they do their own refurbishment in-house, making them among the few who are honest about it. I like OWC and do a lot of business with them but get my refurbed hardware from Apple web-direct.

Could be degraded Capacitors, heat, or dust...

Have you blown the dust out of the intakes lately?

What temperatures does this App report? Especially the GPU.

This App can be used to speed up the fans, all my iMacs would be dead without it.

Never ever ever drop fan speeds below the speed that the Mac itself is calling for.

MacFansControl…

https://www.crystalidea.com/macs-fan-control

OK, great, the temperature does not seem to be the problem.

As far as dust, look at the back of the moc for a round exhaust port for evidence of fur bunnies, might try a vacuum on that grille... Apple made iMacs incredibly tough to blow the dust out.