It's pretty hard to tell how the redstone is connected from your video but I think I've got it working properly by connecting the signal to the middle instead of the end and turning one of the repeaters around since, if I was understanding your connections properly it was delaying the wrong pistons.
In your design the redstone power is added and removed to/from one side first, then there's a delay and it gets to the middle, then there's another delay and finally it gets to the other side.
In my rearrangement it starts in the middle, then there's a delay and it reaches both sides.
It's the power being removed from the first side before the middle that breaks your build.
Pistons can't push/pull pistons that are extended, when the power is removed from the outside of the first side the outside pistons try to pull the inside pistons but the inside ones are still extended and so don't get pulled. And the door blocks on the first side are still pushed out so they can't be pulled by the outside pistons either. Then, after the delay, the inside pistons lose their power and pull the door blocks on that side in, that's why that side of the door is left half open.
The two halves of the door are independent of each other so the delay between them doesn't matter.
The second side works because the delay goes in the right direction.
Both sides close properly because the delay then is built into the way the wool blocks power the pistons and the repeater doesn't matter.
The wool with the redstone dust gets powered and powers the pistons below it when the redstone dust is powered, the wool under the repeaters doesn't get powered.
So even though the inside wool gets powered before the second outside pistons the inside pistons aren't there yet so they don't get powered, so the outside pistons push the inside pistons and door blocks into into the middle and then the inside pistons get powered, by being under powered wool, and push the door blocks into place.
But when you try to open the door, by removing the power, the repeater delay has to go in the right direction.
BTW, I got a kick out of being reminded of the SCP Foundation!
I had entirely forgotten about them but the SCP in your icon seemed familiar somehow and worried away at the back of my brain until it clicked.
I put a switch there in the same spot in your screenshot above and I seem to have gotten it working. I didn't try that before because the way I had it set up, it made sense to put it off to the side (at least in my opinion). Works like a charm now. Now time to hide the switch somewhere..
Every time I've done it, I have really enjoyed working with redstone, it's always so interesting to me. Just need to learn more about it and get the hang of it and I'll be in business.
And I'm glad you got a kick out of the reminder of the SCP Foundation, 999 is one of my personal favorites.