Hello and welcome to the forum.
Try this as a test.....
At the sign-in screen, don't sign in.
Use the power icon on the bottom of the screen, while holding the shift key and choosing restart.
Navigate through the Troubleshooting -> Advanced Troubleshooting screens until you arrive at the screen that provides "Startup settings"
Click that and when you restart, choose #5 Safe Mode with Networking.
Ignore any messages that may pop up reporting something like this app can't start / run and ignore the fact that everything looks big and ugly, it's using a default VGA driver for video.
A lot of Apps won't work, but just to test your start button, click it and scroll down to Windows Accessories and click Internet Explorer and see if it works and will launch the browser.
After a minute or two just use the start button to choose restart. See if the start button now works in normal boot.