Maybe, but I think the concept of metro areas doesn't help much when trying to define people moving from small towns to cities.

For example, the Philadelphia metro area has over 6 million residents, while the city itself has only about 1.6. Problem is, the metro area includes extremely rural and small towns within about a 1-2 hour drive of the center of the city [1]. I'm not sure I'd consider someone living in exurban/rural Chester or Montgomery counties, 50 miles from the city, to be living in Philly.

[1] https://en.wikipedia.org/wiki/Delaware_Valley

Using the population within city limits to measure, rank, or even describe a city isn't very helpful because city limits are completely arbitrary. Metro areas, while still arbitrary are much less so. Take Atlanta--everyone who lives here considers Atlanta to be anything inside the perimeter (I-285), and the urban population is about 4.5 million, but the city limits only cover half of that area (~130 square miles) and include only about 400k people. Then you have Sitka, Alaska that covers almost 3k square miles, while the urban center is only 2 square miles.

>rural and small towns within about a 1-2 hour drive of the center of the city

According to this, the vast majority of the metro area is within 1 hour.

https://app.traveltimeplatform.com/#/search/0_lng=-75.16843&...

>I'm not sure I'd consider someone living in exurban/rural Chester or Montgomery counties, 50 miles from the city, to be living in Philly.

The number of people living in rural areas is small compared to the people living in closer more densely populated areas. Subtract 10-20% from the population numbers to account for them and you still have a number more useful and less arbitrary than population within city limits.