Good idea to replace failing hard drive on Late 2013 iMac with External SSD?

Yes you can do that. You could put a bootable clone of your current system on it with either CarbonCopyCloner or SuperDuper and then use that. Make sure you format it Mac OS Extended Journaled and GUID partition scheme to make it bootable and delete any software that was on it.

I would suggest purchasing an enclosure and then a separate

SSD rather than one of the "integrated" solutions.

As mentioned, OWC would be a good source for an enclosure as well

as SSD, though they tend to be a bit pricier.

I have had very good look with the "better" Startech enclosures (not the cheap ones).

As far as SSDs in those enclosures, I have had very good luck with Sandisk, Toshiba,

and Kingston. I was actually running my late 2013 iMac off a 1 TB Sandisk Ultra II

as a boot drive for a long time and just using the internal HDD as a clone backup.

Once my AppleCare ran out, decided to take the chance and opened up my

iMac to put the SSD internal.

I would stay away from cheap SSDs (or hard drives) - you get what you pay for. I prefer buying mine here:

https://eshop.macsales.com/shop/storage

They are Mac centric and have good customer service.