27 inch iMac late 2013 only one resolution option

Try holding the Alt (option) key when you click on the Scaled button.

If that doesn't help?

Reinstall the macOS, as per > How to reinstall macOS - Apple Support

If that doesn't help?

Then perhaps someone used an inferior Display panel, cable or GPU for a repair.

I don't know why it's doing it – only what the behavior should be. The only times that I set the resolution on my 27" Late 2009 iMac to something other than 2560x1440 – just to play with it – I quickly changed it back.

What happens if you install Catalina on a spare external drive and then boot from the external startup drive? It could be that there is some file messed up in your user preferences or in a system library (maybe the previous owner was playing in Terminal with something they didn't understand?), and a clean installation would fix it.

How to download and install macOS - Apple Support

If it did, then once you got all of your applications and data installed on the spare drive as well, you could clone that drive back to the internal one (using Carbon Copy Cloner or SuperDuper!). (Even just copying key display-related configuration files back might do it, but then you'd need to know which ones to copy. I don't remember that, offhand.)

Note: I would recommend that you have at least one bootable drive on hand at all times. Do not erase your only startup drive, and count on Internet Recovery to reinstall an old version of macOS. I have seen reports in these forums of people who did that, but then were stuck when reinstalling from the Internet didn't work as expected.

So it was a refurbished iMac from Apple, right? Then it would have been prepared properly. It's a long shot but give this a try: boot into Safe Mode according to How to use safe mode on your Mac and test to see if the problem persists. Reboot normally and test again.

NOTE 1: Safe Mode boot can take up to 3 - 5 minutes as it's doing the following; 

• Verifies your startup disk and attempts to repair directory issues, if needed

• Loads only required kernel extensions (prevents 3rd party kernel/extensions from loading)

• Prevents Startup Items and Login Items from opening automatically

• Disables user-installed fonts 

• Deletes font caches, kernel cache, and other system cache files

NOTE 2: if you have a wireless keyboard with rechargeable batteries connect it with its charging cable before booting into Safe Mode. This makes it act as a wired keyboard as will insure a successful boot into Safe Mode.

Go to Apple > System Preferences > Displays > click on the Scaled button.

A 27" Late 2013 iMac has a native resolution of 2560x1440 pixels. Something's wrong if it isn't offering that to you.

A setting of 1280x720 would make text very large, and I'm guessing it's a non-Retina 1280x720 (low resolution) one based on your statement that things seem pixelated. In Retina "like 1280x720" mode, the letters would be big – but the letter shapes would be drawn at a 2560x1440 level of detail.

Where did you get the iMac?

Did you get this screen the first time you turned on the iMac?

If you didn't it what did you get?

If the hello screen didn't show it wasn't prepared for resale properly. It should have been prepared according to this Apple document: What to do before you sell, give away, or trade in your Mac - Apple Support  

Hi, I'm also having this issue, same mac and same macOS, did you find a fix at all?

thanks!

It was brought from Apple in 2013, owned it since new and has been faultless right up until now

Sorry I thought this was reply was to another thread of mine, I’m not the OP so ignore my comments lol sorry!

Been there, done that! 🤣