I've had 5 25-06's over the past 45 years...3 store bought and 2 I built...but I don't have one now...
I used 75 gr Hornady Vmax(after they became availible, Noslers and Sierras before that) at 3500fs plus for varmints, 100 gr Noslers and Sierras at ~3300fs and 120 Nosler, Hornady and Sierras at ~3100ft...and all my rifles would put 5 rounds into a dime at 100 yds with prepped ammo.
I have two 7-08's right now...a single shot XP100 15" bbl and a 24" switch barrel for a Ruger receiver...both I built...the XP is a varmint killer par exellance, but so is the Ruger with the same load but takes deer just as quick with 154-160 gr bullets. 120 gr Noslers, chronographed, do 2600fs plus in the XP and ~3100fs in the 24" bbl, 160 gr Noslers do ~2700 in the 24"...I can get ~3000 out of a 280 or 284 in a 24" bbl without straining anything...and a bit more if I use all my tricks, but there is no need so why push.
I've also had two 280's... bought one and built one...I never felt undergunned and it killed everything I pointed it at...maybe not as quick as the 338 and above cals, but just as dead.
I like larger calibers and heavier bullets and have many rifles in the 338 and over calibers so I wouldn't use ANY of those questioned right now for anything larger than a Muley, but I've taken Elk with all them...120 gr with the 25 and 160 gr with the 7mm's....all the Elk I aimed at ended up as steaks, roasts and hamburger.
Ballistics tables don't really tell you anything but speed and energy and are biased in many ways for many reasons...and for me it gets tiresome when that "stuff" gets waved about to prove "something", but if you don't have experience then you have to use something to base a decision on.
The 25 uses a lot of powder because of the larger case...so does the 280...the 7-08 uses less powder and is more efficient, but you can't compare a 7-08 to a 280 OR a 25 because you can ALWAYS get more velocity out of the larger case than the smaller one...and therefore more energy.
It would be better to compare the 25-06 to the 25 Souper or 257 Roberts than the 7-08 and the 7-08 against the 280 or 284.
As far as the Handi goes, I would pick the 280 over the 25-06 any day because they both have the same barrel length and base capacity, and I can shoot 100 gr to 190 gr bullets in the 280 and cover ALL the bases as far as game is concerned, so the potential for a better, quicker kill is much higher in the 280...but that depends on the ability of the "shootee" rather than the "shooter".
But it really depends on WHAT you want the rifle for....for varminting I wouldn't go over 6mm, for deer/antelope the 7-08 is great, but for all around and for one rifle I would pick a 35 cal, Whelen or 358 Win and save for a 243/7-08 barrel for the lighter stuff.
Luck