Having to start the browser in private mode is great but how about starting in a specific profile. This way I can keep my personal profile vs work and school separate. I think this would be a big win.
Each profile should keep their sessions separate so cookies and such.
You can specify the profile to launch with by appending
to the launch command. If there's only one profile, its directory is called Default. Subsequent profile directories are called Profile 1, Profile 2 etc.
A typical desktop shortcut, therefore, would have as its target
"C:\Program Files\Microsoft\Edge\Application\msedge.exe" --profile-directory="Profile 1"