Is the Canon 580 EX simply not compatible with the Yongnuo YN622C-TX controller?
If you have 580EX flashes, they are not compatible with remote manual power control using the YN622-TX. Manual power can only be changed on the flash.
If you have 580EX II flashes, they are fully compatible with remote manual power control using the YN622-TX.
From the Flash Havoc page for the YN622-TX:
Full Remote Control
Flashes that will support remote flash control through the camera menu.
Manual – Full Remote Manual Power Control in 3 groups from the camera menu.
ETTL – Full ETT Ratio Control and FEC in 3 groups from the camera menu.
Canon – 600EX(RT), 580EX II, 430EX II, 320EX, 270EX II
Flashes that do not support remote manual flash control through the camera menu.
Manual – You must set the manual power level directly on the flash.
ETTL – Global FEC control is available from the camera. Ratios may be available, though individual FEC can also be set directly on each flash to achieve a ratio if needed.
Canon – 580EX, 550EX, 430EX
Beyond that, the flash needs to be set to use its hot foot for communication with the camera.
I've tried setting the flash to Master, Slave and Off.
If you set the flash to 'Slave', it will ignore anything attached to the hot foot and will monitor its optical sensor for optical pulses from an optical master flash (or ST-E2 flash controller). You should set the flash to 'Manual' power mode in the same way you would if the flash were attached to your camera's hot shoe. As far as the flash is concerned, the YN622C receiver you have attached to its hot foot is the camera.
The rest of the answer below assumes you are using 580EX II flashes:
When you change a setting via the YN622-TX controls (or the camera's menu), you need to do a half press of the camera's shutter button to tell the YN622-TX to transmit the change to the remote flash if it has been more than a few seconds since the TX has communicated to the flash. As far as I can tell, this is a power saving feature as the 'wake' and 'set' signals are combined in one very short transmission.
Similarly with regard to 'Auto' zoom settings, when you move the zoom ring on your lens and do a shutter half-press the head on the flash moves and the connection stays active for 2-3 seconds during which time the zoom head on the flash will move with a change in the position of the lens' zoom ring. After that 2-3 seconds have elapsed another half press needs to be done to sync the flash's zoom setting to any changed position of the zoom ring on the lens.
Don't forget that the 580EX II must be set via the flash's control panel to 'default' settings for power and zoom or the YN622-TX can not change the setting on the flash. The flash should be set to 1/1 manual power (or ±0 EC in TTL mode) and "Auto" zoom (There will be an "A" to the left of the displayed zoom head position) using the control panel on the flash in order to allow the YN622-TX to change those settings. If the flash is set to another power level (or EC setting in TTL) or a specific zoom setting (There will be an 'M' to the left of the displayed zoom head position) via the flash's control panel the YN622-TX will not override those settings. Also note that pulling out the diffuser panel on the flash is manually setting the zoom to 'wide' and the zoom head will not move regardless of the setting made via the YN622N-TX. You're also locked out of changing the zoom setting on the 580EX II's control panel when the diffuser panel is pulled out.
The behavior you describe in your question would be exactly what one should expect if the the last power setting applied using the 580EX II's own control panel was not 1/1 and the last zoom setting applied using the 580EX II's own control panel was anything other than 'Auto'.
I've also seen situations where specific older firmware versions of the Yongnuo YN685 (a flash very much like the 580EX in some ways) and newer firmware versions of the YN622C-TX lead to the display on the YN685 not accurately displaying the current settings entered via the YN622C-TX. When I take the picture, though, the settings displayed on the YN622C-TX are applied. When I set the power to 1/128 via the YN622C-TX and/or the camera's menu and the flash's panel still shows 1/1 it's pretty obvious when I take a picture I am only getting 1/128 power as opposed to full power. The power level indication on the YN685's control panel only changes after the flash has been fired.
You may want to test to see if the 580EX II is doing the same thing: power level is being changed, but it just isn't showing on the flash's control panel until the flash is fired.