I downloaded SPSS Statistics 26 from the JourneyEd link my university sent me, and it downloaded just fine on my Windows 10 laptop. None of the dialog boxes that popped up during the installation process reported any errors. However, when I go to actually run the program, the most that happens is the icon pops up on the taskbar for a second then disappears, nothing else. When I right-click to run the program as administrator, I am asked if I want to allow the app to make changes to my device. I click "allow", but then the icon shows up on the taskbar for a second then disappears like before. The licensing wizard installs just fine and I am able to run through all of its prompts without issues.

I have tried uninstalling and reinstalling the program, trying both the 32-bit and 64-bit installations despite confirming my laptop is 64-bit, and explicitly running every step of the download/installation process as administrator despite being the administrator already. How do I make this work?

I had the exact same issue. I went into my file explorer and opened an SPSS file. Then I pinned it to my taskbar.

Pinned SPSS that is.