Here's some things you might not have already tried. Make sure the account you're logged into windows with has Administrator Access. It's not always clear when the operating system hooks are reset so it's best to reboot after making any changes and testing or observing if that had any affect. This can be tedious so I understand most people won't persist but if you want the results bad enough then imo it's worth it. Though I can't count the times I've spent way too much time on something and still failed, those occurrences usually lead to some positive idea or insight, though often not until much later, sometimes months later.
With all that in mind, please accept my apology in advance, if some of this is too basic, because indeed it's often the simple changes, that if untried, were all that are necessary from the get go.
If open IE, click on Tools, Internet Options, then switch to the General tab, you can then click on the Tabs settings. When The window with the Tabbed settings opens, there's an option to RESTORE DEFAULTS. You probably want to take a picture of the current settings first. My reasoning for this is so you can confirm Restore Defaults actually changed something. IOW, if all the settings on that page were already at the Default level then Restore Defaults wouldn't necessarily overwrite the old settings. Thus imo, it's worth the effort to change all those settings to something different first, reboot, and then (for the first time) try clicking Restore Defaults. Although this might not make sense to many people there are obscure ways software avoids wasting effort when it thinks it's unnecessary and this is exactly the type of thing that can make it difficult and tedious to figure out why something is not being reset, even after you told it to reset. In this case you want to be sure it responds and actually overwrites all the settings.
If nothing else other than tenacity has been voiced thus far than again please accept my apology for being so wordy, and not being able to help. Given enough time, imo you can beat this and you have to start somewhere. Another thing you can try is press down and hold the Control and Shift keys while clicking on a link embedded in a web page as this should force it to open that link in a new tab and switch to that tab. Though you've already said that doesn't work in the other modes, some obscure mode such as this might be enough to force a different route through the software which again eventually, one of these ways has to reveal some information, give a different result. If you follow this path and pay attention to the smallest differences, I assure you this is something someone somewhere can resolve.
