I've been shooting the 6mm's for well over a decade, my first rifle was in .243 which took several of my first heads of game. The bullet was the run of the mill cup & core 100 grain soft points. They work just fine.
on the other hand, I will also tell you that I don't use those bullets for game anymore. There was a comment somewhere above stating "wasting money on premium bullets". I will tell you flat out that is some of the purest BS ive ever heard. The bullet is the cheapest part of any hunt, from prairie rats to Moose. Does a high volume shooter need to shoot Barnes or GS? Of course not, but that is a specific scenario. Same as the elk hunter using cup & core. Not a good choice IMHO.
Testing, evaluating & identifying a one size fits all projectile is going to be a challenge. Which is why I shoot 3 different 6mm's from .243 to 6-284, all with different bullets suited for a specific task.
Can it be done? Sure, as most of the posters here prove, but
there is ALWAYS a trade off, you have to give this to get that.
I've had great luck with the 87grn V-Max in my 243 for the light stuff up to Antelope. I use the 107 SMK in my 6-284 for the longer range stuff. I was shooting the 85grn Partition in my 6mmAI quite frankly because it shoots in the .3's. After shooting a couple critters with the light Partition at high velocity... I am switching bullets.. think landmine.
I intend to use the 80grn Barnes TTSX with the AI at about 3600fps.
We'll see how that works out.
If you really want to use the same bullet for everything, I think the 85-95grn area is going to be where you want (the 105 A-Max should be on your test list). I recommend sticking to the heavier end of the spectrum if I were intending to shoot Deer & not worry too much about the varmints. You can predict drop, but drift is a whole 'nother story.