Angel grace is a power source but it’s also what “neurologically” connects the true form to the vessel. There’s a non-grace connection, but the grace is like the nervous system. Or at least the electricity in the nerves.
When you remove an angel’s grace, they don’t lose their true form, they just lose the ability to sense it and control it. It’s like if you severed the nerves in your arm, you now can’t feel your arm, and you can’t make it move.
The reason that other angels’ grace doesn’t work is because the grace is tuned to each specific angel. The wrong “frequency” burns up the angel slowly because their form wasn’t built to handle that “frequency” of grace.
Also, when an angel takes a vessel, their consciousness is sort of… moved forward to the vessel. Angels do have their own “brain,” their own separate consciousness center within their true forms, but for the sake of being able to properly control the vessel on the 3d plane that it exists on, the center of consciousness is relocated to the vessel. So when the grace is removed, the angel is trapped in the vessel, rather than removed from the vessel and stuck in their graceless immobile true form.