About the topic of anchor fingers, I made a little video where I talk about the various approaches I've observed trad whistle and flute players using:
https://www.youtube.com/watch?v=CJ_ypnajB44&t=6s
About approaches to playing C natural, I've always done both the half-holing and the cross fingering. Indeed the two approaches aren't mutually exclusive, and on the uilleann pipes especially are commonly done in conjunction. I've always done the same on whistle.
Passages that go high B > high C natural > high B I've always done by leaving the top finger down and cracking it open a bit.
For middle C natural (which I call that because middle D is its neighbour, and the flute can play low C natural) I've always done the crossfingered C natural, with or without also venting the top finger.
Yes there are players, very good players, who don't use the crossfingered C natural. But be aware that most of these players aren't playing half-holed C naturals in all the spots that you think they would. Rather, they tend to play C# in many places that, in theory, a C natural would be played.
The most extreme example of that I've personally observed is when I arrived at a house party to find two young Irish fluteplayers sitting in front of the fire-place hammering out reel after reel, and nary a C natural between them! They were excellent players and knew loads of tunes.
In many tunes it mattered little, but it was odd to hear tunes like Rakish Paddy, the Gravel Walk, the Nine Points of Roguery, and the Bank Of Ireland played with C sharps throughout.
There are many tunes that would be extremely clumsy to play if you tried half-holing all the C naturals. Of course there are many ways of restructuring tunes to avoid difficult notes, and Irish pipers and whistle players have long done such things regarding the note F natural. There are many tunes that I've heard played with F naturals throughout by fiddle, banjo, and box players that I've heard with F sharps throughout by pipers, fluters, and whistlers. I've heard players who only half-hole C natural apply the same sorts of stratagems to the note C natural. (F natural has to be half-holed on keyless uilleann chanters, flutes, and whistles.)
One cool thing about crossfingered C natural is that it's part of a shared suite of performance practices amongst trad Irish woodwinds (uilleann pipes, flute, whistles). Obviously this isn't a practical argument for using crossfingered C natural, but it does help explain why it exists. Add to that the fact that the Baroque flute used crossfingered C natural as well.