This is quite important.
Early 19th century chemists found that
molar specific heats did appear about the same for all atomic solids. This is called the Dulong-Petit law. They justified this to themselves by arguments as or even more vague than yours
And it was damned useful to them. Extrapolating it allowed to give atomic masses to elements. Note that the law doesn't need to be exactly true for this purpose, approximately true will serve as well.
Then there were other properties - the 'colligative' properties - osmotic pressure, freezing point depression etc. that could be used to estimate atomic and molecular masses too.
Now here I am not very sure of the history, but as far as I know they all started as purely empirical laws. They are known by names of discoverers like Raoult, van't Hoff etc. I think they just used them without much explaining them, and I guess they just thought the Dulong-Petit law had the same status. (I remember at school we were only given these empirical laws with no theoretical explanations and we just did the calculations with them without questioning, like the laws were just good luck that solved a problem that otherwise you could see no way to crack - and I guess the early chemists were the same). As far as I know the colligative properties were only rationalised by Gibbs by the 3/4 century, I just looked up, and he was clear that specific heat was not one of them or was anomalous. Anyway I am not sure that his (formal, mathematical, macroscopic at that stage) work seeped into chemistry very fast.
Rather later there was a full molecular kinetic explanation like what you sketch. But, in brief, after the initial usefulness the law and its explanation didn't work very well and even had failures so clamorous as to call for a whole new theory of dynamics! Successes using some quantum assumptions, first by Einstein, and I think others on gasses, had a major influence (the Jeans report) persuading scientists to take these assumptions seriously. If I am not mistaken more than the black body and photoelectricity by themselves. I would guess that this is because you can talk about specific heats in purely mechanical terms without getting into the already hard to understand electromagnetism.
As you see I am hazy on the exact history of this issue, I had been meaning to ask in the History section about histories, books, about how the ideas re colligative properties etc. evolved.
What I meant to say is you'll meet up with the questions you raise at least twice in your studies, once in questions of atomic and molecular weight determination, and again in basic quantum mechanics.