Would you place the mic at the mouthpiece or the end of the flute?
Flautists love to sway around as they play; it’s rare to see a flautist not do that. It causes uneven volume and, when they move past the mic, phase problems.
I used to use a small diaphragm condenser mic and ask them not to move. That makes musicians nervous, impedes their performance, and they do it anyway.
The best solution I’ve found is a mic intended for flute. It attaches to the flute at the mouthpiece end, which is where you want it, and stays in place no matter how much movement there is. The one I use is an Audix ADX10-FLP, for both for live performance and recording.
Since the sound emanates from the mouthpiece itself, it makes sense to have that close to the mic. Although the end furthest away has a hole, the sound doesn't come out from there. Very simple to put one's ear in certain places while a flute is being played, and the answer is there for all to hear.