Your second video makes it look like you have enough functioning to get into the menus. Run through the display test and see what happens. Also take note of the memory test as you go into the menus.

How's your 5V at the CPU board? Looking at the schematic I think you can check it across C10.

Which one are you calling the Dallas chip?

In your second video, the CPU LED appears to be flashing but the Display driver LED is on solid. I think the problem may be on your display driver at this point.

I think we each read the schematic a little differently. You're describing the use of the DS1210 as a supervisor chip, which is outlined as Section B in the data sheet. I see it operating as outlined in Section A, as a battery backup to finish write cycles to avoid corrupting the memory, and also preventing write cycles if the power supply voltage is too low. In either case the DS1210 is powered through VCC when it's available, and not just when the battery is installed. I believe the role of the supervisor chip is being played by U14B and it's combined inputs.

I don't know a whole lot about the Display Controller, but I suspect what you're seeing is the Control Board is losing communication with the Display Controller, and is reseting the Display Controller as described under the theory of operation. The part that I'm confused by, is why the Display Controller works normally as long as the battery prevents full system boot-up. My guess is that part of your Display Controller ROM is corrupt, and the Display CPU doesn't lock up until it accesses that ROM location. I'll be interested to see what solves the problem.

Good luck!