By the way, I solved this:
THe external drive contained an existing backup to the same computer called *.sparsebundle. Those are only made when a drive is being accessed by Time Machine over the network. It was there from before I moved it to attach directly through USB.
TM was able to access that sparse bundle without errors and update the backups there which is why my backups were running correctly.
However for whatever reason, when I try to enter TM to see the backups it did "nothing". No messages, windows, or errors of any kind, including in the system logs.
The solution was to delete that sparse bundle and do a new backup with the drive directly connected via USB. This created a folder called *.backudb as expected, and now I can do backups there and I can see the backups by entering time machine (taking me to the time travel window.)
Ouch. I say the fact that this happened without any error message is a time machine bug although probably not many people hit it. At least give an error, but better, just work with the .sparsebundle, why not?
Remark: you can still access the backup data by opening the backup via the Finder. This doesn't give you the fancy star spangled background, but you can at least access your files.