Thanks to the generous and timely help of @mei in the Firefox Desktop Community chat room the issue was found to be related to a bug that was apparently "fixed", but in the process caused the issue that I posted above. Very sad day indeed!

I'm not going to mark this as solved, because technically it isn't, in hopes that somebody will see it and maybe figure out a way to return this very useful feature back to Firefox without causing a regression in that bug fix.

Hold Ctrl+Shift while clicking the link.

or

Hold Shift while middle-clicking the link.

Thanks @zeroknight, I can confirm that Ctrl+Shift/Click does work but there seems to be a bug for me; if I Ctrl+Click a link at any point on the DuckDuckGo page, then try to Ctrl+Shift/Click after, it highlights a bunch of text instead of opening a new tab.

Shift+Middle-Click does not work for me on my linux box, but that could be an unrelated issue. It does seem to work on windows however.

Honestly though I still can't call this a solution as both of those options are more difficult to actually perform as apposed to the easier option of Ctrl+Click which was the working option before.

My hope is that maybe somebody will bring the Ctrl+Click option back to auto switch and maybe set Shift+Ctrl/Click to open in the background....

But thanks again @zeroknight! I really appreciate it!

Note that holding Shift toggles the 'open in background' setting as set via "Settings -> General -> Tabs" or most cases of a middle-click, in some cases you can the right-click context menu (Shift + right-click) in case JavaScript on the webpage interferes what more likely is if you left-click.

Hi @cor-el, thanks for that info, but I'm a little confused by your post. Can you confirm which option is the 'open in background' setting? Are you referring to the 'open in tabs instead of windows'?

After closing the DuckDuckGo tab and clearing cookies I was able to see the Shift+Middle-Click did open a new tab and switched to it automatically in my linux box. So @zeroknight is correct, there are two options available as far as I can tell that do the same thing as what Ctrl+Click used to do. However the bug I mentioned earlier with Ctrl+Shift/Click is still present and for whatever reason, the shift+middle-click option actually wasn't working at first, but again that could be unrelated.

I know this is probably something that isn't going to get fixed since there is at least one fully functional other option available, but it was just so freaking fast and easy to open new links when browsing with ctrl+click in that text area, this is going to be a highly missed feature!

Another disappointing result in this change is that even if you select the option in "Settings -> General -> Tabs" to "When you open a link, image or media in a new tab, switch to it immediately" it still does not switch if you Ctrl+click in the "text area" of the DuckDuckGo results beneath the link. The tab still opens only in the background unless you click directly on the link itself.