I bought a battle rifle in the day, the civilian HK91. Battle rifles aren't necessarily select fire, I'm surprised the Moisin Nagant owners haven't raised an outcry.
In general, as there is no official definition, battle rifles were mostly the full power cartridge semi auto self loading designs, generally including the M1 Garand, G43, SKS, M14, FNFAL, G3, etc. If anything, as a class, they were largely found too lightweight for full auto fire.
Assault rifles follow the grandaddy of the all, the STurmGewehr 44, which set the pattern - full auto magazine fed intermediate caliber, usually less than 18" barrel, and designed to optimize two factors limiting human operators - too much recoil, and an inability to see targets and hit them beyond 300m. If you reduce the recoil, the operator will shoot it more, and more bullet flying will mean more hits - even by random chance. It makes no difference if it's aimed or not, a bullet striking a soldier tends to limit their response, and incapacitation - the inability to fire back - is what counts.
As for the "overloads their medical logistics" argument, that hasn't worked out so much. Much of the fighting since assault weapons have been adopted has been against adversaries with very little logistical depth. Many don't even have medical corpsmen in the field.
Main battle rifles, whether or not bolt action, aren't assault weapons. It's the specific layout of features that counts, not what caliber the bullet happens to be. If anything, it's been a logistics issue that the adopting nation already had bullet machines running and simply couldn't afford to change over to a smaller, more aerodynamic shape, which would give better results - higher speeds and flatter trajectories. That was a strategic tradeoff, as the battlefield condition of not being to engage targets accurately beyond 300m made either choice equally useful. The bullet makers "won," although they actually lost, not being able to tool up and start other designs. That's the amount of weight logistics can bear on war - you accept compromises to get something in the soldiers hands, rather than a lot of nothing.
Same thinking goes to why we buy an $1100 combat weapon, not a $3000 one. No real effective difference for the money.