Based on my admittedly limited experience, I've found that while copper bullets certainly kill whatever you hit with them, they don't do it quite as quickly as lead core bullets of similar weight. Quick disclaimer: I hunt almost exclusively with handguns, but I have used plenty of rifles and rifle cartridge-chambered T/C Encores to take down deer. The bullet type that drops deer the quickest (barring a spinal shot, of course) is a broadside hit in the lower chest cavity with a medium weight polymer-tipped bullet at moderate to high velocity. A 150 grain SST out of a 300 Weatherby, a 130 grain Nosler Ballistic Tip out of a 270 Winchester, and a 155 grain A-Max out of a 300 Win Mag Encore pistol, all broadside shots, had the exact same effect: that deer dropping in its tracks. That's not to say they all did, but several did, and the ones that didn't drop didn't go too far. Copper bullets, specifically the 150 grain TTSX I tried in the 300 Encore? The only one that dropped in its tracks was a spine shot. I don't think the spitzer shape lends itself well to rapid expansion that is needed for a quick kill, but the trade-off is deeper penetration than you would achieve with a same-weight lead core bullet. On that note, if you want to try out a lead-free bullet, you might consider using variants at the lighter end of the weight range for your cartridge.
On the handgun side of things, I have some experience using vastly different copper bullets with some interesting results. My primary hunting handguns have been 460 S&Ws - a Performance Center Model 460, and a T/C Encore. Using the Model 460, I used a handloaded 200 grain Barnes XPB - the spitzer version designed for the 460 - pushed along at 2,500 fps to take a coyote and a whitetail doe at 40 and 100 yards, respectively. You'd expect those big 45 caliber bullets to have knocked them to the ground, but the reality was both ran off with little indication they were even hit. The coyote ran about 60 to 70 yards before falling over, and the doe made it to the trees well over 100 yards away before going down.
However, contrary to my previous experience, this past season gave me some interesting insight. The first buck I killed, a massive 9 point during muzzleloader season, was unfortunately not dead when I approached him. I quickly drew and fired my CVA Optima pistol, hitting him with a standard 45 Colt designed 200 grain Barnes XPB at 1550 fps and killing him. Then, less than a week later, my 460 Encore dropped another buck in his tracks with a 200 grain Monoflex - another bluntly shaped pistol bullet - from the Hornady Handgun Hunter factory ammo line.
TL;DR: No, I do not think spitzer copper bullets are better for deer hunting. However, traditionally shaped copper pistol bullets definitely are.