I note in the datasheet, page 8, that they draw some 413ma of power. Each. That’s 826ma if you use a pair, such as in the SeqV3. I note:
“Restriction: the 7805 gets very hot when it delivers currents above 500 mA, so only core modules without backlit display like MIDIO128, MIDImerger, MIDIfilter should be supplied over this port.” -ucapps, CORE page.
This requires a separate power supply for the displays?
Maybe. Especially as you will have your LED’s on top of that. But you want to use the same transformer. Otherwise, you can use a more powerful regulator, and/or a heatsink (search for that here, you’ll find good info). You’ll have to figure out the maximum that your system could draw. Add up all the LEDs and backlights, add a bit (few hundred at least) for the rest, and add a bit to be safe. There is a 3A version of the 7*05’s (search those too). I personally wouldn’t recommend a completely separate transformer for the LEDs, you could have issues with different grounds…
mbseq_v3_interconnections.pdf simply shows two LCDs wired directly to the CORE. I’ll suppose that a separate power source must be used for the LCDs, and the CORE only drives other PCBs without LCDs, as stated. I’ve searched the three sites and don’t see an answer.
This will work in some cases, and not in others. It really depends on the parts you choose. The schematic you saw is the simple way to do it with non-fancy-schmantzy parts. 
It Would probably work in your case too… the 7805 should go up to 1.5A (search it, you’ll find more posts like yours), but if you do that, you will need a heatsink for sure.
The SEQ page says, regarding hardware requirements:
PSU: Wall adapter (Power supply) 9V / 500 mA.
Smash TV’s excellent site says regarding CORE power:
“500 mA is recommented, especially if a backlit display is used”
Am I missing something?
Nope. Your display sucks a lot of juice, and you have 2 of them.
Wiring the transformer raw 8V or so source to both the CORE and a separate 5V regulator for the displays?
8V might be a bit shy. 9-10 is nice, especially if you have a heatsink in case that’s too much. Any more than that is overkill, over 12V is bad.
Just my recommendations, nothing official 