I finally called Ring support a few days ago. Per their support there is no way to do this on the web site. Maybe it will be there in the future since I do still see a faint light gray Delete button that doesn’t function. However in the Android app it does work but you have to use a certain series of screens to get there. Hitting the History button from the default page or Dashboard will not give you the option to delete events.
From the default page or Dashboard of the Ring app hit the three bars in the upper left and select in the drop down menu Devices. In my case I only have my doorbell so I select that one.
-In the next screen select Event History to present a list of Recent Activity.
-Slide one of the events to the right and you will see all of the events will now have a checkbox to the left side of the screen. Select the check boxes you want to delete and hit the trash can on the bottom left. I think you can also hit Delete All on the bottom right if you want to delete everything. You will get a confirmation dialog to confirm you want to delete them.
Once done all other people that have access to the Ring devices will also have those events removed from their list. I think only the owner of the account can delete events.