The .444 case is 0.27" shorter than the .30-06 and it operates a a substantially lower pressure (44,000 CUP vs 50,000 CUP), so a .44-06 could give a substantial increase in performance over the .444. The .30-40 is only a tiny bit longer than the .444 Marlin, so no advantage - plus it has an even bigger rim. Getting a rimmed case to feed reliably in a bolt rifle is tough - two of the few that work are the .220 Swift and the .225 Winchester, both of which have tiny rims.
The .240 Weatherby might be an option, but its case diameter is marginal for a .430" bullet. That case does feed well in bolt rifles and headspaces off the belt. I knew two Alaskan hunters who have .416s built on the .240 case - they worked well and were .400 Whelen performance clones without the problems.
Not everyone is sold on the idea of using cast bullets on the large game the .44-06 is appropriate for....in fact very few are. There are not many appropriate jacketed bullets available in this bore size.