Once again, the operative word is BORE. Dat be the hole in the barrel. The word "Big" refers to size of said "bore"

.30 is considered "small".

The CASE used for the .300 mag is rather large in relation to its bore, not the largest, but kinda big. There are bigger.

Yes, it is powerful and capable of long range. I've owned and hunted with .300 Winmag rifles and have a lot of respect for their accuracy and power. It is a versatile round.

I think you'd find your .300 WinMag more destructive to deer and similar sized animals than either the .458 Lott or the 416 Rigby loaded with the usual bullets used in those rounds. You should try them some time or another on deer, just be advised of their penetrative properties. You'd be surprised at the lack of destruction.