The 30-30 or the 35 Remington are great in the Marlin 336's. In the Late 1970's I bought a Marlin 336C in the 35 Remington and when my father looked at it and saw how inexpensive it was, and the quality, he immediately bought one in the 30-30 a week later. Both performed very well and the recoil is very moderate on both. Both the 30-30 and the 35 Remington is about the same class of power, but the bigger diameter of the 35 Remington has a little advantage attached to the effectiveness but not much.
That being said, I do not know how much you want to spend. If I wanted to reload for optimization, I would prefer a lever action that has a Rotary Bolt and clip fed. Both the Browning BLR and the Henry in the Long Ranger version are a good choice. The next statement is for the Henry. I have not investigated it yet but they are offering a side load tubular version, probably to reduce cost in the 30-30 that is a new model as of this year. That one Might not be a Rotary bolt and something I would investigate before buying, If I decided on a rotary bolt for optimization of reloading. I have not picked up a Henry, but I know if you get the short action such as a .308 W, 6.5MM Creedmoor, or a .243 W, the throw on the lever is much shorter which is advantageous for anyone and especially for a young child. The Marlin lever actions will have a longer throw on the lever than the short action Browning BLR. After being used to the Marlin, I had a friend that had shown me his Browning BLR in a .308 and all I could say is wow, when the lever was thrown. He did not have to take his thumb off the stock to fully actuate the action and really smooth. Also keep in mind if reloading with ones face on the stock the short action is fairly comfortable, but when one goes to the standard action such as a 30-06 the bolt comes closer to the face, while reloading a round into the chamber and one has to be more cognizant of not having ones face too close to the back of the bolt.
Best regard and do what makes sense for the budget and bringing your children along into hunting.