There is nothing whatsoever incompatible between ancient peoples migrating across the Bering Strait during the last Ice Age, and migrations in earlier times. Simply because modes of travel would have been primitive does not mean travel was impossible, either on foot or by primitive rafts or boats. It doesn't take much to make a large, if crude raft.
Even today, the Bering Strait is only about 58 miles wide, and the two Diomede Islands are almost exactly in the center of the strait. It's about 27 miles of open water from Siberia to Big Diomede Island, about 2 miles from there to Little Diomede Island, and then about 27 miles to Alaksa.
Homo sapiens has always been a migratory species, despite the rise of the nation-state (Egypt) and the concept of borders about 5000 years ago.
I'll concede that point, although even a 27 mile journey by raft in open ocean, assuming you could direct it in a straight path, is no small challenge. I would pay money to see anyone attempt it today, using nothing that wasn't available 40,000 years ago.
There has been some reports that mitochondrial evidence (something way beyond my expertise) suggests a south Pacific influx into South America about 15,000 years ago. That seems inherently implausible to me because it suggests an ocean passage far greater than a Bering Sea transit would be, and I don't think ocean currents could explain it.