If Havilah were the same place as mentioned later, where is the Pishon
River? This river remains unknown.
If Cush were the same place as mentioned later, where is the Gihon? This
river remains unknown.
The post flood Tigris runs through the heart of ancient Asshur, not to the
east of Asshur.
The Euphrates is not associated with any region, yet in the post flood
world, it is the river of mighty Nimrod's Babylon.
The post-flood people were likely named for pre-flood people of note. So
post-flood areas named after post-flood people could also have pre-flood
namesakes. Post-flood Asshur may have named the river running through his
kingdom the Tigris after the river which ran BY the pre-flood region named
after his namesake.
The Euphrates, who's headwaters rise from the mountains of Ararat, may have
been named by Noah and family as one of the first rivers they encountered.
I don't see any compelling evidence that Gen. 2 is describing the
Post-Flood world.
Allen