All three of the calibers you mentioned will get the job done very well on average sized whitetails at the range you mentioned. If versatility for bigger deer and other big game is a deciding factor, go ahead and X out the .243. I have one, love it, and have taken several decent sized for NC deer with it, but I wouldnt take it hunting to other parts of the country that have much bigger whitetails, nor would I think of hunting mulies or elk or bear with it. So that leaves the 7-08 and .270. For your uses at under 250 yards, you will probably never notice the difference between the two and neither will the deer. If you ever went on a hunt where a longer range shot was a possibility, say 500 yards, the .270 would have a slight advantage. So for versatility, my vote goes to the .270, but just barely. The price and availability of ammo for both calibers will be cheaper and more plentiful by comparison than most other centerfire rifle calibers out there. Around here .270 is a little more plentiful but your hometown may vary. You mentioned recoil. The .270 will recoil the most of the three calibers but as you said you experienced, it wasnt that bad. The 7-08 is a little lighter kick and if thats an issue for you it might be a factor to help you decide.
I said all that to say this:
I think you should get the .270 unless extra recoil bothers you. If thats the case then the 7mm-08 will be ideal for you
"I don't understand all that I know"