Aspartame does not increase insulin, unless I'm not updated on the current scientific consensus.
Also your idea is very american centric, diet sodas are mainstream around the world and most of those countries did not follow the us into the obesity epidemic.
Not at all an expert, but the brief research says that it does cause insulin creation, though due to it not being actual sugar and also due to that there is 200x less Aspartame in mass in your drinks (as it's much stronger per weight unit), then this spike is small and the organism can easily handle it.
Also your idea is very american centric, diet sodas are mainstream around the world and most of those countries did not follow the us into the obesity epidemic.