Going to take a stab, even though I’m more well versed on politics and political history.

So going back, first, the capital of a city was typically established when that state was granted admittance into the union. When that happened, the largest cities now, typically were not the largest cities then. Take Nevada, whose largest city is Las Vegas, which wasn’t even incorporated until 1905, well after the state was established and Carson City named its capital. Typically, the rationales for choosing a capital are pretty straightforward, In my above example for Carson City, it was chosen due to the high volume of people due to the Comstock Lode silver strike in the 1800’s, and the schmoozing of the man in charge as well as it having been home to a large part of the states economic activity. It could house the legislature, had hotels for business men visiting to talk to legislators and the government, etc. It was also where a ton of transportation into California passed through. But per the wiki page on Carson City, it came down to one man choosing that it be the place, and at that time was the most populated city in the state.

There are different reasons for every state. Take New York, which seems like a no brained for NYC, until you realize that NYC is just one city in a HUGE state, typically in those times the capital was chosen on proximity to access for the rest of the state, they were typically central, and had routes crossing through them that would act as throughways to the rest of the state. Contrast tht with California which chose Sacramento due to both proximity and the economic activity of the gold rush… while Los Angeles became the economic hub of the state, that didn’t start to happen until Hollywood, a d really didn’t take off until after WW2’s population explosion due to the interstate system and Los Angeles’ strategic position between San Diego and San Francisco. citation

I guess it just comes down to a lot more than the population, although plenty of states have capitals with the largest population centers, and that’s usually because the states business, economy, and commerce hace all been thriving in that city since it joined the union. Like Boston which was an economic powerhouse during the founding and still is today. Also, keep in mind the revelations brought by the war of 1812 which saw the Capital of the US burned down, and think how that would fare in a state tht had both its governmental sphere and economic sphere centrally located in one place. It may not apply as much today, but in 1800 if a country attacked a state capital, it’s economy would be unaffected (relatively) allowing for the state to continue to function. Now if it’s economic center is crushed, then government can still guide if they’re not located together as well (which is part of the rationale behind the fact that DC is separate from any state).

On New York: Why isn't NYC the capital of New York? written by u/discovering_NYC