I mean, sure, there are benefits to those larger cartridges. Really depends on your use case. ELR Light or Tac class (if you can get the AXSR under 19lbs), I think the 30 cal with high BC 230s at 3000+ fps is superior to any of the 338 cartridges, especially if you can get into the 3200 fps realm.
For the AXSR, you can have a 1.350x5" straight taper to 1.1 at 30", you can still be underweight for the ELR light. I also run the TBAC 338 Ultra SR 100% of the time and run Ckyepods DP, which is a significant portion of weight right there.
Another reason I like the 300 NM is you can mag feed the heavies, whereas 338 LM you'll neuter the cartridge if you try to mag feed the 300gr bullets, or run into donut issues down the line. And 33/37XC is going to be strictly single-feed.
If you're shooting Heavy class, I think the AXSR will be handicapped compared to the other purpose-built rifles.
For the AXSR, I'd stick with 300NM or 300 PRC, but run long barrels for the PRC to make up for the velocity. This is just theoretical, but the hit percentages increase drastically at extended distances (2000+ yards) with 230s at 3100+ fps compared to 2900fps or the 338 cal 300s at 2900 fps.