It rings very true to me. I believe that people who have trouble sleeping are often people who didn't learn that much that day, and conversely, people who learn on a given day seem to need more sleep.
I noticed this with my kids. A few months before they started learning to talk, say, at about 13 months, they seemed to need more sleep, even though they weren't exercising any more. This is the period that you can really begin to see them starting to "get it", about a lot of things in their world. Even though they aren't talking yet, they are beginning to understand what you are saying, so they are learning fast.
13 months has to be within a month or so of the median age for starting to walk (in my country at least, I know some start much younger) but you say your kids didn't increase their exercise; presumably then they were already walking at that age?
We have 2 boys that have been treated almost identically, obviously there have been differences and only one of them is a first child. However they have so far had vastly different sleep patterns. The one who sleeps least appears to have developed fastest in linguistic terms. His brother, the slower, was certainly no slouch (and without wishing to boast) is pretty close to top of his class for reading despite being one of the youngest.
I meet quite a lot of different parents & babies/toddlers from different backgrounds and find there are a lot of variations in sleep pattern - I don't have anywhere near enough contact with them to establish correlations with development and sleep pattern though.
My kids could be outliers. I'm sure such studies must have been done?!
tl;dr anecdotes are anecdotal and my experience doesn't bear out your hypothesis
I find that I need a pretty consistent 8 hours of sleep a night for normal operation. However, I've found that when I'm doing something challenging (say, a startup or a new project at work, or learning a new programming framework), this spikes upwards fast. I need about 9.5 hours when doing something moderately challenging, and up to 11 hours if it's a problem that requires my absolute full attention.
I remember reading somewhere a hypothesis that sleep exists so that your mind can convert short term working memory into long-term memory. Naturally, you would need more of this if your short-term working memory is more filled up with complicated stuff. But this goes at odds with the cultural practices at Caltech or MIT or med school - if it's true, you should structure the curriculum so that students work very hard during the day, but then have enough time to go home and sleep it off before learning the next day's lesson. I wonder if there's any research on this.
I find I also dream very vividly when working on a hard problem, but this could just be correlation, as REM cycles increase in frequency toward the end of the night, and if the night is extended, there ought to be more of them.