Certainly an interesting topic, and the answers thus far are informative and just a bit baffling to me.
On vintage flutes I’ve tried, oxo xxx works quite well for the second octave C. (I mean the C above middle.)
I have several keyless flutes by modern makers, plus two keyed flutes (Olwell “Pratten” and McGee “Rudall Perfected”).
On every one of these, oxo xxx is around 25 cents sharp sans adjustments. I can’t work well with that.
That fingering is almost always fine on the third octave C. But the keyed C is easier for me there ~80% of the time.
On the Olwell, oxx ooo comes out surprisingly strong.
Yes Jem, I know what you mean and this is usually more muffled but he’s made it work well here.
It is also nearly in tune, just a tad sharp of the tuner meaning about right in many contexts.
But oxo xxx is unsuable to me, and oxx xox is disturbingly flat, and only a bit better in tone.
On the McGee, oxx xox is dead on and quite strong. Only slightly more muffled than the keyed C.
However, oxx ooo is quite noticably weaker than on the Olwell. And oxo xxx is also unsuable.
(By the way, the oxx xox and oxx xoo is pretty much the same flutewise. Former makes easier transitions mostly.)
On both Olwell & McGee, the keyed C is the strongest and most in tune.
Not to belabor the obvious, but quick transitional notes are a different matter.
We seem to be talking about sustained and exposed notes here.
Interestingly, the Olwell and McGee require different (keyless) fingerings to get the C# in tune in the second and third octaves. Probably because of the compromises needed for different approaches to the C nat.
So in short, yes (groan) you have to do some things differently on different flutes.
About thumbholes…a tangential subject…
I’ve now put these into about six keyless flutes (or flute sections), maybe four Olwell bamboos.
Also over a dozen whistles including my Burkes and Copelands, a Goldie and an Overton.
I was a doubter initially and slid into conviction. I can’t see any downside and lots of plusses.
If you play a Boehm, there’s no learning curve - same fingering only no key, just a hole.
One side benefit is the ease of bringing C# into tune on both registers with the same fingering.