No. You fly the same heading, which on a globe puts you in the South Pacific.
Do you understand what a heading is?
Perhaps you should try drawing out just what you think the path should be on a FE, and mark the heading along various points.
As Portland is further north than Miami, a compass heading will have you continue to move towards the north. You would eventually reach the north pole, the magnetic north pole if you were using a magnetic compass without correcting for magnetic declination, the geographic north pole if you were using a gyrocompass.
But you now bring up another point about FE maps. I was stationed in Thailand. On the way there, we started at San Francisco, to HI to Guam to PI then into Thailand. On the way back it was Bangkok to Hong Kong to Japan to San Fransico. If the Earth is flat it would have been shorter and faster to fly to over Canada, Siberia, China, and into Thailand
As you went via Guam, it is quite clear that you weren't going the shortest route. By any chance was this during the cold war, when it likely wouldn't have been smart to have US planes flying over Russia and China?
But your claims of the best route is also wrong. If we ignore that it is heavily dependent upon which FE map you use, and just use the common NP centred AEP, you don't go over Canada at all.
And the shortest route for a globe and FE are quite similar.
Both start going north, head over or near Alaska, and go over Russia and China.
The only difference is that the FE route goes more into Alaska and Russia, and the FE route may also go over Mongolia.