The point of the X-1 has always been that it was the first human piloted plane that broke the sound barrier in level, controlled, and repeatable flight.
Aerospace engineer and amateur WWII historian here: I'm not aware of any credible claims from that era. The only WWII-era claims to have broken the sound barrier (by piloted craft) I've seen can be attributed to poor pitot tube design which gives erroneously high indicated airspeed readings when the ram pressure gets high enough.
A reconnaissance Spitfire I believe is what got closest, at something like .9 of Mach measured by a fancy pitot system at the RAF test establishment.
Shock waves develop inside the pitot tube under certain conditions, and that causes a spike (or sometimes a drop) in pressure which is interpreted by the instrument as a corresponding rise or drop in airspeed.
Edit: as for the 262 claim itself it was pretty handily debunked in a post-war NACA technical note. If you dig around on the NTRS you can find it.
https://en.wikipedia.org/wiki/Hans_Guido_Mutke
In other words, it's the same problem that the piston-engined aircraft of the time had: Anomalous readings at high speed.
For an airplane to reach Mach 1, this mainly requires thin wings and enough power/dive. To control it, you need a flying tail, and swept wings and "waisting" the fuselage reduces the amount of power needed.
So almost all of the planes in WW2 were "too thick" and incapable of supersonic flight, though could have exceeded Mach 1 with modified wings, tail and in a dive.
The US P-80 (1945) and F-86 (1949) jets had underpowered engines, in the sense that all early jet engines had limited power.
However the F-86 had swept wings, and later versions of the F-86 had a flying tail, so if you pointed it at the ground, you could get a sonic boom without losing control.
Source: I study aerodynamics.
Creating a sonic boom in a deep and dangerous dive is an interesting quirk in a fighter, but it doesn’t set new speed records that would interest the general population.
The book "Aces Wild" [1] says that it was unofficially broken before that in an XP-86 (The experimental F-86, a jet powered fighter) in a dive, and witnesses on the ground heard sonic booms before Yeager's flight that day. But history favors the hero's, and in this case, the science: We know 100% for sure, because of science, that Yeager officially broke the sound barrier first.
It is quite possible that the F-86 test pilots who stuck their planes in a dive likely did it first. It is well documented that the F-86 was capable of Mach 1 in a dive. Even so, that's not what the history books say, so that's not what we teach kids in school. I suspect it'll always be the same with the F14 computer vs the Intel 4004.
[1] https://www.airspacemag.com/history-of-flight/mach-1-whoduni...