I had the exact same problem where the initial health warning screen loaded with the flashing click A button to continue type message. After pressing the A button a black screen would just hang and the system menu would not load.
The following steps fixed my problem:
1: Unplugged the PSU, RCA, and sensor bar cables from the Wii.
2: Unplugged the PSU from the wall.
3: Removed SD card, Gamecube memory cards, and Gamecube controllers from Wii.
4: Let everything sit in this state over night.
5: Plugged PSU into wall socket
6: Plugged RCA cable into Wii
7: Plugged PSU into Wii
8: Powered up the Wii with the Wiimote power button (not the power button on Wii)
This resulted in the Wii booting to the system menu screen. I then plugged the sensor bar in and the SD card. Wii is currently still working.
Following the steps above, if I tried to boot the Wii using the power button in the Wii, the problem persisted. I was only able to boot to the system menu using the power button on the Wiimote.
Maybe this will work for others with this issue.