The devices that aren't showing up aren't managed by apps. They're headphones and speakers. It seems I may have been in the wrong part of the settings app, as I've just now discovered that the area of settings labeled "Bluetooth" shows the device name and an option to pair a new device but doesn't show the currently paired devices. I thought my device name was just hidden.
There's a second separate settings page for "Pair new device" that shows currently connected devices and has the same option to pair a new device as the Bluetooth page. It doesn't let you control the current device's name, but it does have a weird link to a general connection preferences page that includes a link to the first Bluetooth one, as well as links to similar pages for NFC, Android Beam, Cast, Printing, Received files, and Chromebook.
I had a lot of trouble finding both of these pages, and the fact that these are two different settings pages does not help at all. They've really made this a lot more difficult and confusing in Android Pie.
There's a second separate settings page for "Pair new device" that shows currently connected devices and has the same option to pair a new device as the Bluetooth page. It doesn't let you control the current device's name, but it does have a weird link to a general connection preferences page that includes a link to the first Bluetooth one, as well as links to similar pages for NFC, Android Beam, Cast, Printing, Received files, and Chromebook.
I had a lot of trouble finding both of these pages, and the fact that these are two different settings pages does not help at all. They've really made this a lot more difficult and confusing in Android Pie.