Which kicks harder? Who cares, the recoil of either can be ignored by a seasoned marksman, yet might prove to be a bit much for a beginner, particularly if he or she has a preconceived fear of it. Choose the rifle that is the most suitable to the end use, chambered for which ever cartridge it is available in at the time you purchase it. Don't wait a month to get a .270 that is available in a similar rifle but chambered for a different cartridge today. There isn't enough difference between any of these cartridges to give one a great advantage over another. Usually a short action .308 will be slightly smaller and lighter than the standard length .270 or .30/06, although there are exceptions. If your rifle of choice doesn't have a good recoil pad installed at the factory, that needs to be a consideration. If the rifle needs to have the LOP shortened, you can upgrade the recoil pad at the same time.
The answer to your recoil dilemma is handloading, rather than attempting to choose a milder factory round. Load low recoil ammo until such time as your family can handle more. A 125-130 gr bullet loaded to 2600 produces mild recoil from any cartridge from 6.5 to .308, yet still produces suitable terminal ballistics for hunting deer sized game. My, small for his age, 12 year old grandson did pretty well shooting my wife's cut down .30/06 loaded to this level last summer. This rifle has a ghost ring and post rather than glass, a 21" light contour barrel, a 12" LOP, and weighs about 6.5 pounds . The only 130s I could find were TTSXs so his training and practice was on the costly side, but the smile on his face made it all worth while. Of all the guns he shot while here, this was his favorite, which trumped shooting a scoped .243 prone from the bi-pod, which he expected to like better. He also shot my .375 Ultra with 300 gr cast bullets loaded to 1200 . . . and hit the steel at 100! Not only did he get to shoot (lots), but he was able to help load his ammo as well.
The only factory ammo he shot while here was .22 rimfire, 9mm and .45 auto, and a few 12 ga field loads. He did better than you might expect with the full size .45, but had to lock his arms so the 1911 would cycle. My 9mm is a 3" prohib that is difficult to hit with, with and my disdain for it as a target gun quickly rubbed off on him. I knew the 12 ga was too much for him, but he insisted on trying it. I think the problem was the shotgun's LOP was much too long for him, but the clubby little sawed off Winchester 37 was the least of the evils, as I don't have much in the way of shotguns. I should have borrowed a decent 20 ga for him to try on some clay birds.