I've shot a few deer with both, most within 50yds. I've never seen any advantage with the .44, if one exists, even at that range. Past that the .30-30 looks better fast, and drops deer with more authority. At closer ranges, I've had the best terminal performance with Sierra 210gr HP bullets. Most exit on a double lung shot, but quartering shots, even at closer ranges, not. 300gr bullets penetrate well, but expansion is minimal. I've never seen a deer drop on the spot with a double lung or heart shot with either round, unless a shoulder is hit. Wasted meat.
Shot placement is everything, until penetration falls off. The longer bullet in the .30-30 will shoot through more deer stuff most of the time, although the 265gr Hornady FP is pretty good at that too.
Not sure how many deer I've shot with a rifle, maybe 75 or so. Everything from a .243 to .375 in that mix. Far more with a shotgun, at least a hundred of those on crop damage permits. Another couple hundred with a bow. Quite a few with a black powder rifle and .44 bullets in a Sabot, velocities are the same as a .44M rifle. The .44's are certainly effective within their limits, devastating, not really. On average, a well placed arrow is usually better at getting deer down than a .44 or .30-30. The .357 in a rifle, devastating? I gotta wonder how. Even foxes and coyotes can run a bit with a double lung "hide shot". I'd use one if I was hungry and had nothing else, but otherwise, not.