We should not forget that the support class is able to heal or resupply themselves. In practice this allows them at times to be even more aggressive than assault players. This further emphasizes your point that support is expected to be up close and should have SMGs available to them. At the same time it allows them to hold a more defensive role where they "suppress" the enemy by large volumes of fire. Having infinite ammo, certainly helps that role. Which also makes a case for them having access to LMGs.
This may sound controversial, but when it comes to engineers, I wouldn't necessarily say that they are always in long range engagements. They're at best at medium range when engaging a tank or perhaps they're even closer if they looking to use C4. LMGs don't necessarily shine in chaotic combined arms situations as they in prior titles often lacked the ADS speed to excel in CQB fights. They are best used to when you can get into an advantageous position first.
It's almost as if you'd want to give the engineer the assault rifle, as that gives them the necessary flexibility to operate anywhere in between short, to medium-long ranges. It would also mean that the SMG weapon class should be much more pronounced as a CQB powerhouse. The Carbine weapon class could have better hipfire/average firerate than assault rifles, but more damage drop off at range.
So you could for example play with a different setup:
Assault: SMG, Shotgun (Has weapon sling to compensate for lack of range on larger maps)
Support: Carbine, LMG
Engineer: Assault rifle, DMR (Fast firing)
Recon: DMR (Slow firing), Sniper Rifle
Assault not having access to assault rifles makes even less sense.
