"The moment could not be better!"
Translation:Der Moment könnte nicht besser sein!
I checked with a native speaker and she basically confirmed your statement:
"Der Zeitpunkt könnte ..." is the best way to put it,
"Der Augenblick könnte..." is OK,
"Der Moment könnte..." sounds weird to her and she would never use it.
So why aren't neither Zeitpunkt nor "Augenblick" accepted alternatives?
Not the same. As I understand it. The difference is in the time frame. Like in the sentence. This will only take a moment. (such as waiting for your dinner to arrive.) As opposed to "It happened in a moment (Augenblick) and poof it was gone." The time it takes to blink your eye.