Except that it could be cold and icy. Yes, if there is snow in the mountain passes, you will be required to have chains. That is OK because your daughter will want the chains if she ever has to travel to the coast when there is fresh snow in the pass. Buy some nice ones that are 'easy' to put on, take off.

Snow tires for your trip sound good, but the Interstates are almost always clear. And driving on bare pavement at 70-80mph for 2,000 miles might wear them down a bit. Except for northern Indiana, close to Chicago, where it can snow a foot/hour. And when that happens, you might just want to spend the night somewhere warm or follow a snow plow.

I used to spend Christmas in CT, and drove from Chicago. I would leave CT at around 9pm so the kid would be asleep. Only once did I have to follow a semi through western PA where the snow was over a foot deep. No snow tires, no chains. No snow plows. No traffic, until Cleveland. That trip took 14 hours.

There is no cruise control allowed when the pavement is wet or icy, you right leg may get a bit sore, so count on a few extra stops to get the circulation going again.

Four days. Five if the weather is bad. This is a BIG country.
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Keeping it running is better than buying new