So I've been using Skyrim and SKSE for a number of years. I recently reformatted my PC with Windows 10. Then I started rebuilding my Skyrim mod list. SKSE is installed correctly and launched through its shortcut. Unfortunately every time I try to start a new game I get complaints that SKSE is not loaded. The main menu does not show the customary SKSE print near the bottom denoting version. When I look at the skse_loader and skse everything looks as expected. I don't have Skyrim installed in the program files directory. That being said I'm a little bit at a loss how to troubleshoot this.

Check your security programs maybe they are blocking it from running.

Are you still starting SKSE using the old method? If not I would try loading Skyrim through the SKSE Loader and see what happens.

That could cause your SKSE issue. If you haven't done that already then reinstall http://www.gamesas.com/topic/1568922-wipz-skyrim-script-extender-skse/ via a SKSE Installer.exe file and doing that will create a working Skyrim - SKSE shortcut on Windows desktop.

All I have is Windows defender and Malwarebytes Anti-Malware Notifications I've checked the logs and don't see anything interfering.

I have launched through the shortcut on the desktop. And I've already tried reinstalling it both a manual method and installer.

Try running the loader with administrative privileges.