I have Handi's in each of the calibers you list. I don't have much range time with the 7-08 or .308, so I don't yet know what the "pet" loads are for each. Both have produced accuracy in the 2" range at 100 yards using Remington Core Loct 140's with the 7mm and Winchester Power Point 150's with the .308. I think I may have fired a half of a box through each.
I've always been very partial to the .30-06 for all the typical reasons. Availability of ammo, variety of factory loaded ammo, versatility, etc. All of those reasons could be said of the .308, as well. The .30-06 does have an edge over the .308 with heavier bullets.
That said, I have been real impressed with the Handi .280's. I currently have three. One is a RMEF model, which I've not fired. The other two have been the most inherently accurate Handi's I've owned. I'm set up to reload, but have yet to do so. I've only fired Remington Core Loct 150's and a few 140's. Each will shoot the 150's under an inch, 3 shot groups at 100 yards. I recently had a problem with a barrel lock-up on one, but NEF quickly made it right by replacing the barrel. That rifle picked back up where it left off. No tuning has been performed on either, including the "O" ring. I use a Simmons 44 Mag on one and a Nikon Pro Staff on the other. So far, I'm very pleased with each.
It took a little work, but my Handi .30-06 is now shooting very well. It always shot well enough to hunt with, but I know what the .'06 cartridge is capable of and wanted a little better. By the way, Federal Power Shocks, Hi Shoks and Fusion's, each with 180 gr. bullets, shoot best in my Handi '06. The Core Locts shoot well, just not as well as the Federal's. I have a Bushnell Banner mounted on the .30-06 and have been pleased with that scope as well.
If I had to pick one of the three you listed, it would be the .30-06.
Thanks.
EVOC ONE