Why does an iMac 27" (late 2013) no longer work as a second display?

You are playing with dead technology. Apple started the deprecation of Target Display Mode in late 2014 and threw the last shovel of dirt in its face in early 2020.

The support article linked above is confusing, poorly organized and, being archived, is not being upgraded to reflect the current situation and requirements, which are:

VIDEO SOURCE COMPUTER (the one sending the video signal to an external monitor):

• A 2019 or older Intel Mac running macOS 10.15 Catalina or older.
• Apple Silicon Macs are not supported
• mac OS11 Big Sur is not supported.

TARGET COMPUTER (the one to be an external monitor):

• An iMac 27-inch (Late 2009 and Mid 2010)
• any iMac from Mid 2011 to Mid 2014.
• The target computer must run no higher macOS than OS 10.13.X High Sierra.

Yes, you could downgrade the iMac to 10.13 (not sure if partitioning works) but the MBP 2019 cannot be running BIg Sur. You would need to downgrade it to no higher than Catalina.

Robin, there are a couple of 3rd Party Apps that you can use extend/mirror your display to another mac. Duet Display and Luna Display.

I'm currently using Duet display between iMac 2012 on Mojave and 2020 iMac on Big Sur. The connection is wireless.

I hope this helps.

Paul

Hi there Robin the wedding photographer,

Thanks for using Apple Support Communities. It sounds like you want to use your Mac in Target Display Mode. That requires macOS High Sierra or earlier. If your Mac is using a later version it would not work.

These articles might help:

Use your iMac as a display with target display mode

Use external monitors with your Mac

We hope this helps!

Robin the wedding photographer wrote:

Was it for security reasons?

Thank you, Robin,

We are end users like you and Apple seldom shares reasons why they do or do not do something.

I think it was more social media than security. We saw on 1-2 questions a month about TDM before last March, after the final blow to TDM was delivered. It was a dead issue. Then COVID came and we now see a dozen or more inquires a day here.

I honestly think some starving web journalist, in trying to cash in on the demand for work-from-home articles, threw something together without researching the subject. Then non-facts got spread around on social media and other info-outlets with less than gold-standard journalism practices, as we have seen in recent examples of political post-processed cattle fodder.

Sounds like a fun profession you have. I consider myself a "semi-professional photographer." That means I almost got paid once!