I've came across two issues that are related. I thought I'd share my findings and see if anyone else experiences the same thing.
One. VoiceOver Preferences does not save the website setting of an activity. To reproduce, go into VO Utility. Create a new activity for a specific website. Go to that website and notice that all works as expected. Return to VO Utility and save your VO Preferences. Reset VO to default. Finally, load the preferences you previously saved and try to go to your activity website. You'll notice the activity doesn't switch automatically. You must manually select it with VO+X. If you then launch VO Utility and examine the activity, you'll find that the website you specified is no longer present.
Two. Command+Tab doesn't announce the name of the first app icon. This seems to occur when you have an activity that is tied to a specific website. While viewing the website in Safari, with the activity in use, Command+Tab away and notice the first app icon name isn't announced. Get rid of all activities, or get rid of the activity website settings, and Command+Tab will behave correctly.
The second issue is related to the first, in that the first bug will clear the activity website setting, causing the second bug to go away.
I have not used activities until now. Except, I do seem to recall I tried them when this feature first became available. And if I remember correctly, I experienced the same kind of behavior and stopped using them. If my memory is correct, these issues have been around for several MacOS releases with no change or improvement.