Hairy...
You're right! Probably more elk and moose taken with either a 30.30 or .303 Brit over the last century than any other caliber. In Scandinavia, the most popular caliber for moose reportedly is the 6.5x55 Swede.
That said, modern bullets definitely improve one's odds of avoiding a wounded, and potentially lost, animal. Where deer sized game is concerned, not so much of an issue. But a bullet, such as a cup & core or even an interlock, hitting a shoulder bone on an elk or moose sending the jacket one way, the core another, and the critter over the hill into the next county, is another matter entirely. More times than I care to remember, I've found Sierra Gamekings from my 25.06 on deer pretty much pulverized beyond recognition...but still makes the kill. On a deer, which generally ain't all that tough in the big bone dept.
Read a story some time ago in a rifle magazine that it was an incident in which just such a thing happened on a huge bull moose, with an appropriate caliber, and things got somewhat dicey very quickly. The fellow it happened to got a tad POed, and set about to invent a better constructed bullet. This is how the Nosler Partition came into being circa 1946 or so.
The short version here on Wikipedia...
https://en.wikipedia.org/wiki/Nosler