Back in the last century, in the days before rangefinders, moderators, twiddly turrets and Creedmoors I used an estate rifle - a Sako TRGS in 25-06 with a fixed 10x42 S&B scope. It was a superb open hill rifle and I shot a lot of hinds with out to long ranges. It was flat shooting and hit hard.
I did look at getting one as an all round rifle and in particular for use on Roe in farmland where ranges are much less. The high velocity and meat damage put me off.
However in these days of monolithic bullets I think the bullet blow problem is sorted.
Is the 25-06 better than others - yes and no. Its faster with 100gn bullet than the 243. It’s like a 243 with an 80gn bullet but its shooting a 110 to 120gn at the same sorts of speeds.
Better than a 270 - probably not and the 270 can shoot bigger bullets, but if a 243 is sufficient for big deer, a 25-06 certainly is.
The 6.5s work on a different philosophy. They shoot long heavy for calibre bullets at modest velocities, rather than a light for calibre bullet at high velocity.
They cause less mess close up, but retain their energy downrange. Due to lack of velocity, their trajectory is less flat for the first 250 to 300m but then they maintain their velocity down range. Also less affected by wind.
The 6.5x55, 6.5x54 and 7x57 were developed as military cartridges in the late 1800’s where the ability to hit man sized targets at 1000 yards and provide harassing fire further was a requisite.
And they gained a reputation for being an easy to shoot and able to much bigger animals than they had right to.
Modern iterations such as the 6.5 CM, 7-08 etc still have the fast twist and mimic if not exceed the early velocities, but do so with less powder, and thus recoil.
Going back to the OP question, either is not a wrong choice. If you are shooting open ground, want max point blank range and don’t want the faff of ranging etc, then go as fast as you can. Otherwise more choice will be down to the individual rifle and scope setup rather than the cartridge.