There isn't really a wrong choice, but if it was me, I would go with the .30/30. When you want more you could switch up to a modern high velocity number for open country or a powerful medium bore when chase a grizzly into the willows, but harvesting deer in the bush is what the .30/30 does well.
At the range a few days ago I picked up 20 once fired .30/30 cases, I don't come across .35 Remington very often, but I do know there's at least one here. The point is that .30/30 ammo or components are never difficult to source, but the same can't be said for .35 Remington.
From a performance point of view, yes the .35 factory load uses a bullet 30 grs heavier than the .30/30's 170 gr, but the bigger diameter of the .35's bullet means that it requires more mass to penetrate as deeply when the impact velocity is the same. If the bullets are of similar construction the upset will be similar giving the .35 the potential of a larger wound volume, but whether or not this matters on medium game could be argued endlessly.
The handloader has more options in either case and lighter bullets can be driven faster, which could be useful should you get a shot along a power line or lake shore. I would be inclined to load up a .30/30 as 2 shooter (one in the chamber and one in the mag) with 125 gr spitzers at 2600 if there was the likelihood of a longish shot.