In WoT's video, I don't understand why you would remove the "tool" legs with a screwdriver. Why not just pull them out with the vinyl chassis attached and then disconnect them? Once the head is removed there is nothing holding the chassis in.
I was never able to remove any legs with a screwdriver without marking the legs, which is why I use either Hans On or the shoestring method.
The person who made this video does it differently then me...
You can switch several of the steps around...
All you need to do is use the leg tool to pull out the head,
Then the rest can be done in your own order.
So yes, you can pull out the frame first.
If you are marking your klickies, try a smaller screwdriver...
Maybe it is bent at the tip?