I think the answer is pretty clear - it isn’t possible to put these remaining few notes in tune with the hoped for fingerings, at least not without unacceptable losses elsewhere. Not surprising, as I’ve mentioned before, the flute is simply too long for our puny hands.
If we look at the right hand, we see three fairly equally spaced holes, which is fine, as we have three fairly equally spaced fingers. But, and it’s a big but, they cover 4 semitones, not 3. Carte, writing in 1851, attempts to draw our attention to the problems with a drawing in which he showed all the holes (finger and key) as if they were all fingerholes running along the top of the flute:

Look at the right hand section. The first and third hole in that section (giving G & F) are well placed but the 2nd hole (gives F#) should move up-flute (and be smaller), and the 4th hole (gives E) should move downflute a fair way (and be much bigger). We make the holes bigger and smaller to try to get around their wrong positions, but we can only go so far. It’s the limitations on human stretch that give us flat F# and weak E, not any intention by the maker that we should use more complex fingerings. We can improve the flat F# by opening Fnat and Eb, but the fingering charts of the era show that that they preferred the simpler fingerings wherever possible.
(The whole pamphlet is at http://www.mcgee-flutes.com/CarteSketch.htm)
Subtly different problems prevent us getting c# up to pitch. The top hole has to work for c#, c nat and as a vent for middle D. It’s location is also limited by available stretch, but we can’t just make it bigger (like we did with F#). Once it’s more than about 8mm, the middle D vent starts to look more like a B hole! Try blowing the softest middle D you can, and you might just hear the phantom B, or the multiphonic B/D. And the c becomes unbearably sharp, even if played “maximally flat” by using oxx xxo. So again, the laws of physics combined with our puny stretch are the reasons behind flat c#. And again, the c key can help, but was not at the top of the preferred fingerings, but typically around no 3, after the second harmonic of low C# option.
I think it’s pretty clear that the players and makers were well aware of the issues and solutions, and the makers were well aware of the physical limitations and possible mechanical solutions. Mechanical solutions put forward at the time included thumb C key (Boehm & later Pratten), use of keys to relocate holes (Siccama), Brille (Siccama and some others), but these generally didn’t seem to find favour. I think this is very interesting stuff - it seems to be telling us that the players back then, like us, valued simplicity very highly, even at the cost of accuracy. I could easily put some extra keys on to my flute to be able to play perfectly in tune, but I’d rather play adequately in tune without the klutsyness of additional keys, or more complex fingerings involving keys.
Terry