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If 80-grain .223" bullets can win matches at 1000 yards, then 110 to 120-grain .257 bullets can do well at beyond 500 yards. It is true that .257" bullets lack the very high BCs found in 6.5 mm and larger calibers, but the potential for long range shooting is certainly there.
The real reason that the .25-06 isn't spoken of much for very long range shooting is the same reason that the .30-06 is not - the long thin cases lack the ballistic consistency to do best at 600+ yards. Ballistic uniformity is essential or there will be a large vertical dispersion at long ranges. The .30-06 has been completely eclipsed for long range use by the .308. '06 rifles are not shot seriously in long range competition except at nostalga matches.
I've not shot a .25-06 at long range, but I have shot a .257 Weatherby at 600 yards on targets. (Shooting rocks at long range is fun but it shows little about the accuracy potential of the rifle.) I have a target somewhere with a 5-shot group fired at 600 yards of 6.0" fired with Nosler 100-grain Ballistic TIps. That was on a calm day with minimal mirrage off a bipod. The rifle normally shoots 0.75 moa at 100 yards.
Best long range .257" bullets:
Nosler 115 BTip - .453 BC
Speer 120 SPBT - .435
Sierra 115 BTSP - .410
Sierra 100 MKing - .394
Hornady 117 SST - .390
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