I'm not going to say anything here that hasn't already been said. I also believe, oftentimes, people ask questions looking for information to corroborate a conclusion they've already come to. But that's a different discussion for a different day.
IMO, what the 6.5's (specifically the creedmoor) bring to the table is incredible balancing of many different aspects of cartridge development.
1. It was designed, from the ground up, specifically to work with heavy for caliber bullets with high ballistic coefficient and high sectional density. There was already a "6.5" on the market (the .260 Rem), but it wasn't designed specifically to do this from day one. It was twisted wrong, and the case length and shape mean you have to eek into your powder space in order to seat the Heavy for Caliber bullets deeper than is ideal. The Creedmoor fixed some issues that the .260 didn't.
2. It's case shape was designed around the same principles, and the straight wall and 30 degree shoulder help to mitigate issues with case "growth" and simplify reloading processes. I've heard
@Formidilosus say he doesn't trim his 6.5 brass, and it's the case design that means it's safe to do so. It does eventually grow, but not at the rate of a 7 rem mag or 280 or 270.
3. It's a lower recoiling round for it's trajectory. I think it's a common misconception that the 6.5 Creedmoor has a laser-beam like trajectory, and it just doesn't. The rifle I packed in colorado has a MV of 2630 FPS, and at 1000 yards (with a 100 yard zero), I drop about 8 mils. It's not a missile strike at all. My brother shoots a Weatherby Vanguard .270 Winchester that shoots Hornady factory Superformance BLISTERINGLY fast (about 3200 FPS), and he's only got about 6.5 mils of drop at 1k. However, our energy (per the Shooter App) at 1k is within 20 ft-lbs of eachother, and our wind drift is different by about .3 MILS, 10 MPH wind at 90 degrees relative to bullet flight path (advantage 6.5). The difference is the 270, moving almost 600 FPS faster, recoils considerably more to reach those numbers. I'm not talking painful recoil, I'm talking the type of recoil that doesn't move you a mile off target when shooting from a bag or off a pack. People talk about recoil as if it's a painful thing, and at points it can be (6 LB 300 Win Mag, for example). Personally, I'm with you OP, the -06 is where I feel I start to feel a really sharp dropoff in my ability to handle the rifle when it comes to recoil. HOWEVER, I had a .308 (Browning A Bolt II) for awhile and was pushing the 168 AMAX at about 2850 FPS, and even though it didn't hurt, it moved me off target significantly more than my 6.5 Creedmoors do (I've got 3 of them now) and was significantly more difficult to control. Again, not painful, just a significant increase in overall movemement. For their recoil and muzzle velocity, my creeds are extremely efficient and other rounds have to either be loaded hotter or be throated longer or twisted tighter in order to keep up. Again, they were designed from the ground up with those things already in mind.
Bottom line, the 6.5 Creedmoor didn't do any ONE thing that anything else on the market didn't already do when it was released. But it gave us a balanced combination of all those attributes (and many others) that no cartridge had before, and taught us that cartridges could actually be designed properly for inherent accuracy. The 6.5 PRC does the same thing but on a "hotter" level. Lots of the other modern 6.5's apply these same principles in order to increase performance marginally, but then you start getting into issues with barrel life and such. It's not that the 6.5's are necessarily anything magical in and of themselves, it's that several of them have been designed in modern times for our modern understanding of ballistics and performance.
I would argue that some, like the 6.5-300 Weatherby or the 26 Nosler, are just downright stupid and pointless. But that's more a dig on the cartridge design than the 6.5mm bore diameter.
Can you push a 180 GR. ELDM from a 280 AI or a 7 Rem Mag if you're a proficient handloader? How about a 230 Grain ATIP from a 300 Win Mag or a 308? Absolutely, without question. But were those cartridges designed, from the ground up, to do that? Nope, not even a little bit.
The 6.5's aren't magical, they just were the bore diameter that Hornady chose to start with when we started into this renaissance of modern ballistics. If they had come out with the 25 Creedmoor or the 7 Creedmoor at the time, you'd have probably started a thread about "What's magic about the 25's" or something. The things companies learned in developing and designing and building rounds like the 6.5 Creedmoor and PRC lead to rounds like the 300 PRC (applies all the same relative balance points to a cartridge that smears the 300 Win Mag across the pavement), the 6.8 Western, the 7 PRC, and I'm sure many more to come.
Bottom line, 6.5 isn't special in and of itself. The modern cartridges they tend to adorn are what's special.
But hey, I'm just some random guy on the internet...