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As long as you take the right approach, there is no reason you can't be one of the thousands of people who travel from coast to coast, every day, even in winter.

With the details of your plans, I-90 (and/or I-94) is certainly the best choice when factoring in the weather. Its the shortest and most direct, keeping you on the road for the shortest amount of time, and leaving you the most extra time in case there is a storm that slows you down. Its a trip that should take you 5-6 full days of driving in good conditions, so getting there in under 10 shouldn't be a problem.

Going down to I-80 certainly doesn't help you, nor does going even farther south. All of the other options still have a chance of seeing winter weather (plus you'd be dealing with the coastal mountains in addition to the Rockies). The only thing you'd be doing by looking at those other options is significantly increasing the amount of time you need for the trip, and reducing the extra time available if you need to wait for conditions to improve.

All-Season tires should be fine, and if conditions are so bad that you need chains on the interstates, then you shouldn't be out on the roads anyway. Interstates are the top priority to be cleared, and it takes a pretty major weather event for them to be to be in poor condition for more than a day or two.