for most of my stuff i want to keep the history and whatever else around. i also never restart my browser or my machine if i can't avoid it. (restarts happen when i don't want them, and that's when i don't want to loose my current state). so i am still stuck with specific sites that i need a second, cleanable mode for. it's the mode switching that is the issue. switching into incognito mode or to a new browser group is easy enough. so fixing either is the way to go.
In my workflow, when I want "Incognito", I just start a new Chromium session on tmpfs. No switching needed.
so i right-click, open link in private mode, or in a tabgroup.
i am done reading, i close the private window or wipe the group.
with your method i'd have to first start a new browser with that second profile, then copy the link to the new browser to read. that's a lot more work than just selecting an entry from the right-click menu.
But that is just about a second of work: my wrapper executable for chromium is named tb, and I use Dmenu with DWM for X window management, thus in four key presses and one mouse middle click the new browser instance is started and the URI copied: "ALT-P" "t" "b" "Mouse button 2".
https://news.ycombinator.com/item?id=20484845