[RightWhale]

the land bridge is mostly underwater

Even now the local residents go way out on the ice hunting. You can still cross from Alaska to Siberia on foot if you go on the ice, and it's not an easy trip, although you need permits. Another thing is why it would take 1000 years to go from top to bottom of the Americas. If somebody wanted to he could have walked all the way in a few months.

[HamiltonJay]

Sure someone today could walk the americas, they know where they are going and what to expect.... however would a largely nomadic society just decide, hey, lets head south non stop? Not likely. The would expand as population and food and water demanded than simply head south young man...

I think 1000 years to spread out completely across americas is pretty darn fast given this sort of expansion and exploration pattern.