Great comment! You touched on something that got me wondering: do you know of something similar to MGF that determines a distribution but is not based on expected values?
CDFs and characteristic functions determine distributions.
This may be pedantic, but an object that determines a distribution that isn't based on expected values is the CDF. In my first probability course in undergrad I think we defined probability mass functions and probability density functions first and defined CDFs in terms of them, but from a measure theoretic point of view, the CDF is more fundamental since it is defined for continuous and discrete distributions (and also for distributions that are neither).
Of course the expectation value and the probability measure are really two sides of the same coin, the expectation value being equivalent to integration with respect to the probability measure.