Unclenick, you mentioned:
A lot of the barrel time sweet spots are several tens of milliseconds wide. . .
My bad. I thought microseconds and typed milliseconds. I'll go back and fix it. Figure a bullet going 2500 fps will have 33μS/inch of travel. A load that stays accurate for a ±1% of powder charge in a .308 (about ±0.4 grains; not uncommon to find) will run just about ±32μS according to QuickLOAD, so almost a 2" span of barrel length. This explains why, in a post you had in another thread, the Garand with its muzzle funneled back 3/4" might not show any loss of accuracy timing with a good load tuned to a full length of rifling.
If you look at Varmint Al's animated FEA of a rifle firing
you'll see his muzzle doesn't complete any vibrational cycle while the bullet is still in the barrel. The bend resembles what you see when you pull a rope taught and crack the end like a whip to send a transverse wave down its length. The recoil's vertical component pulls an upward bend into roguhly the first two thirds of its length. The muzzle is too far away for the speed of the transverse wave to reach it before the trough following the first bend occurs and turns the corner to swing the muzzle up. Thus it turns the corner at something close to the third harmonic node. It is the recoil moment's application of angular momentum from the receiver that forces this deflection profile, which is why it is dominant, rather than the first harmonic or any other harmonic. It's pure mechanics. Indeed, Varmint Al's conclusion is that the deformations are far more important to muzzle position than any natural harmonic or vibrating mode. It does, however, happen that the bend from this whip crack is right about at the third harmonic node, so you can roughly estimate timing from it. It is important, though, to realize the barrel isn't actually ringing yet, because the string isn't done being plucked.
So the barrel isn't actually ringing at the point the bullet exits in the animation. Instead, the muzzle deflection is only about 90° into what the first cycle of a third harmonic mode vibration would be at the time the bullet exits. If we take your barrel with a 63 Hz fundamental mode, the third harmonic is 189 Hz, which has a periodicity of 5.29 milliseconds (yes, I really do mean milliseconds this time). 1/4 of the way into that first period is a rough approximation of the bullet exit time, so that's at about 1.3 milliseconds. Pretty typical of medium power cartridge barrel times. Other barrels will have other deflection profiles, which accounts, in part, for why tuning them works at all.
Of course, there is also Chris Long's pressure wave theory. It's another form of deformation. I'm still not 100% sold on the idea except to say it does seem empirically to be synchronous with multiple barrel times at which groups shrink. Whether it is a primary or adjunct factor, I can't definitively say.
Gunplummer makes an important point of principle. The whole phenomenon of load tuning is gun and ammo synergy. If you have a bad barrel or one in need of recrowning, have bad bedding, need bolt lug lapping, have loose scope mounts, etcetera, the gun isn't going to shoot well with the best ammo you can possibly build for it. Similarly though, you could take the best benchrest gun ever made and if you load pulled surplus bullets from the Far East, bullets with dinged bases or unevenly thick jackets, fill the case poorly with a powder too slow to ignite consistently, use charge weights that vary all over the map, seat some primers hard while leaving others proud of the case head and everything inbetween, it won't shoot well either. You need the best of both to get the best precision shooting.
I've previously advised people with a new (to them) gun, to pick up a box of a good grade commercial match ammo, like Federal Gold Medal Match, and see what the gun will do with that before they embark on a load development program. If it fires a shotgun pattern with that ammo, they need work on the gun first. On the other hand, if it fires an moa or better, or if it fires any other commercial ammo and moa or better, they have a gun that will likely respond at least some (if not a lot) to load tuning.
The reason for the above is the bigger the initial group, the more difficult any improvement is to see. Group diameters caused by different independent error sources combine as the square root of the sum of the squares of the individual group diameters they would cause in an otherwise perfect gun. This means that an improvement that would improve a 1/2 inch group to a 1/4 inch group will only improve a two inch group by about 1/32 of an inch. Pretty hard to see. The point is, get the big group errors tamed first, then worry about improvements that only do fine tuning.