The 44 magnum is very popular here for pigs, but experience supports 8x57 statement. It is adequate, but only at quite short range and there are better options. The 357 magnum isn't even in the ballpark. In a lever, i'd be looking at the BLR in 358 Win.
The very best rifle i have ever used for this type of shooting is my little Browning Dualis .308 pump. The advantage of a pump action over a lever is that you don't have to move your trigger hand. My rifle is essentially self-loading offhand under recoil, and in all honesty i'd take my little pump anyday in the timber over any semi-auto i ever used, as it has the reloading flexibility of a bolt. The only thing i might prefer is a little SxS double. I don't know if pumps are legal in the UK, but if they are it would be worthwhile trying to locate a BPR/Dualis, especially in .308. The Remington 7600 is OK, but not in the same class as the Browning and particularly the short action Browning.