The 416 Taylor is a better way to go. Especially if you are going to have a rifle built. Much better value for the dollars spent, and the 400gr bullet at 2200-2350 is nice to shoot, kills VERY well, never give performance problems and does everything you would want any 416 to do.

For longer shots a 375H&H is a better all around rifle then a fast 416, because more power then the 375 delivers doesn't really help you in the real world at longer range. So the Weatherby kicks harder for no advantage.
Animals are not dangerous at 300 yards. They have to be able to touch you before they can hurt you.

If you want more power for closer ranges, I would recommend a larger bore and bigger bullet, not more speed.

I believe a 458 Win or 458 Lott are better stoppers at 100 yards and less, and the 375H&H is better at everything 150 yards and more.

The 416 Taylor is between the two.

More pleasant to fire and shoots flatter than a 458, and hits harder then a 375.
A VERY good combination.