Existence of polyaminomethyl compounds
I wouldn't say that they don't exist, but there's very little to keep them from collapsing to an amidine, which makes isolation quite difficult.

This is completely analogous to orthoformic acid, which, as a tetrahedral intermediate, rapidly collapses.

Substitution helps to stabilize the tetrahedral intermediates a lot. For example, both trimethyl orthoformate and tris(dimethylamino)methane are commercially available.

EDIT:
Forgot to address this in my answer, but having t-butyl groups probably makes things worse due to steric hindrance. Neopentyl is bad enough...