I went back and read some more of the old thread where you described the problem of a small physical change to the overlap caused a large frequency shift.
A solution to that is more distance between the plates.
Having 6 sq in overlap with 1/4 inch spacing equals 5.4pf
Having 5 sq in overlap with 1/4 inch spacing equals 4.5pf
So that means sliding one plate 1/2" changed the capacitance by
only 0.9pf. Again I don't know the capacitance needed.
Alas, you still need to figure out a mechanical device to hold the plates firm and make the adjustment.
In theory I think what you propose would work. But in practice, the mechanical precision needed to finely adjust the plate separation is daunting. For a 5m-long, 50mm-wide loop (equivalent conductor diameter assumed to be 20mm) resonant at 21 MHz, the AA5TB calculator says a capacitance of 19.883pF is needed, i.e. 2.892uH loop inductance. Changing the capacitance just by 0.9pF as in your example yields a whopping shift in resonant frequency down to 20.540270 MHz - about 460 kHz or 5 times wider than the predicted loop bandwidth of 91 kHz. (EDIT: whoops - the above calculations are for a 0.25-wavelength circumference loop at 21 MHz. My loop is 0.33-wavelengths long, meaning even more inductance is present and even less resonating capacitance is required.)
So even finer mechanical precision would be needed to adjust the spacing, and that is complicated by the fact that the copper foil I am using is so thin. Also the foil is not perfectly flat (with small twists and wrinkles) so the capacitance change with distance separation might not be linearly controllable, possibly leading to jumping over the resonant peak.
The method I used now (which, actually, is the same as the proposed PCB-based homebrew differential variable capacitor in the other thread) has the advantage that the loop ends are (a) fixed in place relative to one another and (b) have little stray capacitance (since they are flat beside one another). Then the overlaid common strap can be mostly fixed in place with just one tail end left loose and acting as a book capacitor. This allows easy mechanical stability of most of the capacitor bulk (everything is affixed to the common plane of the dielectric) while leaving only a small dangling flap for the fine tuning.
Also the effective capacitance is divided in 2 by the series capacitor arrangement.
Frankly, I'm still a little (pleasantly) surprised that the latter method (flat tails with common overlapping strap) works when the former method (spacing between mutually-facing straps) didn't. Since the latter method works, then it should also have been possible to lay the strap ends flat (eliminating mutually facing surfaces) with a tiny bit of common overlap and eliminate the third common strap . But that didn't work for me - maybe the loop ends in such close proximity already introduced too much capacitance. Probably, with enough precision in construction, any method could be made to work.
Right now the dielectric is a flimsy polyethylene freezer bag. Next, I need to visit the dollar store to find candidate polyethylene-based products that are stiffer (1mm thick) and could be repurposed as the dielectric.