This is the biggest detriment in my mind about using a .243/6mm. Out west, where there is more open land, if your game runs a little before dropping it's no big deal. Hunting in the east or midwest where there is thicker forest, it can be very difficult to track (and possibly lose) game because the entrance and exit holes are small. 6mm caliber will take your game with a well-placed shot, no doubt. Finding it may be more difficult than using a larger caliber.
Would those be desert or Rocky Mountain mulies you're hunting back east?
I don't think a whitetail is a problem with a .243, especially at woods ranges. Even .22-250 is a popular choice here, .243 being acknowledged as the low end of serious deer calibers. A Barnes or a Partition WILL exit. I don't think one needs a magic bullet on whitetail, though. Near 400 lb Rocky Mountain mulies, yeah, I'd want a Barnes. The problem with the gun on mulies is distance. Don't wanna shoot over 300 out there because the little booger is runnin' a little low on juice past that for big mulies. If you use a decent controlled expansion premium bullet, you won't need to sweat the penetration, though inside 300 yards.
I'll admit I don't have a .243. I inherited my grandpa's .257 Roberts when I was in high school in the 60s and it's killed the most deer for me over the years. I've shot 'em with .50 cal minie, 7mm Rem Mag, .308, among others and the little Roberts (not much more gun than .243 with factory loads) had killed 'em just as dead, some biggish ones, too. I hand load it pretty hot, now, but just use a 100 grain Sierra Game King bullet. It's a good penetrator. Only thing that'll keep it from exiting a whitetail is LOTS of bone. I had ONE shot enter the on side shoulder through the scapula, bounced up and shattered a vertebra, bounced down and went through an off side rib before stopping under the skin, only one that didn't penetrate completely. That vertebra made a loud, audible "CRACK" when it shattered, was wondering what did that until I started butchering.
It also explained why he seemed to almost flop before I banged. I have used 117grain bullets that didn't open up well at 3050 fps, Hornady interlock. It went straight through a 160 lb buck like a pencil. That one went 75 yards before piling up at a fence it didn't have juice enough to get over. I was lucky to find it. Only deer in a few dozen that gun has killed in MY hands that didn't die DRT.
I like the 100 grain game king in that gun, 1/2 MOA accurate. It's like surgical precision.
I know it will expand, too. I quit using the interlock. With small calibers, a good bullet is important. The newer premiums from such as Barnes are more guaranteed to penetrate AND expand EVERY time.
All that said, the only Rocky Mountain mulie I've ever taken was with a 7mm Rem Mag across a canyon about 360 yards.