I have spun up a fresh self-hosted site on DO with the chat plugin and I can confirm that this also happened to me:
- Installed using the standard install guide (editing in the Chat plugin to app.yml)
- Registered admin
- Completed setup wizard (first 3 steps)
- Enabled Chat from plugin page
- Added staffadminstrust_level_3trust_level_4
- Refreshed page
 
 
- No Chat bubble on screen
- 
/chat/browseredirects to Latest
Other notes:
- Only having staffin the allowed chat groups also does not make Chat appear for this admin user
 
 
- Making them moderator does enable chat for them
- After granting moderator status and then removing it Chat is still enabled for this admin user
