[Rainbow Demon]

Both Japan and Italy discovered the short comings of the 6.5 cartridges they were using before WW2 even got started. The Italians found the 6.5 did not have enough long range punch when fighting far less sophisticated foes in Africa during its colonial incursions. The Japanese found their Type 30 with round nose bullet was not up to snuff when fighting Russia before WW1, then improved the bullet and cartridge for the Type 38 and again found it lacking in penetration of obstacles when fighting the Chinese.
In both situations the Italians and Japanese defeated their enemies, but in spite of relatively weak small bore cartridges rather than due to them.
Lack of range and/or penetration cost lives that need not have been lost and made their job a bit more difficult.
Both militaries attempted to completely replace the 6.5 with a circa .30. Economic concerns rather than satisfaction with performance led to retaining obsolete weapons, just as the U S retained the M1917 for non combat use and the British continued using the SMLE till supplies of No.4 rifles were available.

In Afghanistan the U S and UK troops have on occasion found themselves out ranged when Taleban opened up from 600+ yards with 7.62X54r rifles and MGs.
Otherwise why the rush to produce 7.62X51 Designated Marksman rifles and reissuing of 7.62 LMGs to suppliment the 5.56 squad autos?

A 6.5-6.8 weapon with the best modern projectiles and propelents may be barely adequate for the conditions that are becoming more and more common, but the .30/7.62 MBR cartridges are already enough gun for the purpose.

The 5.56 has its zone, which is not much more than spitting distance against unarmored foes with a clear shot or very light cover. One could do near as well with an M1 Carbine or PPSH in those conditions.
Effective body armor has all but killed the SMG as a battlefield weapon, and improvements in body armor is on its way to doing the same for the intermediate cartridge assault rifles.
As for vehicles, even some civilian auto and truck bodies can be a tough target for the 5.56, and it doesn't take much to hillbilly armor a vehicle to withstand fire from assault rifles. Even WW1 era light armored cars would be proof against almost all modern intermediate cartridges.

The U S military's Urban fighting technical manuals reveal the shortcomings of the 5.56 very nicely.

PS
To better understand why bullets of less than .30 are a dead end for infantry use, take the .30-06 AP bullet as an example. Weights for these were given as 165-168 gr with full steel core.
The longest and heaviest 6.5 military FMJ bullets of the time were around 156 gr with a lead core. How long would a 168 gr steel core 6.5 bullet have to be, and how far would it have to extend into the powder space to maintain the standard OAL of that cartridge?
The longer an AP core is the more likely it is to break apart if it hits at an angle.

The 7mm ran into its own problems when it was found that you couldn't pack a sufficiently large tracer or incendiary charge in the small diameter bullet for anti aircraft or aerial guns.
Once a bullet diameter reaches a certain lower limit bullet construction has to become more and more exotic to maintain the necessary level of performance.
If depleted uranium bullets were available in 5.56-6.8 they might match the penetration power of the cheaper electronic furnance steel core of .30 AP.