The other Bandit
You'll note that at the end of my previous post I compared the MG-3 to the Stoner-Ares M202 "Medium" class machine gun in SR5 (what I would call a general purpose machine gun). I made no attempt to classify the MG-3 as a light machine gun in SR5 terms, as it is very clearly no such thing.
That is also precisely why I compared the MG-3 machine gun to the G-3 assault rifle, as both weapons fire 7.62mm standard NATO rounds; with the MG-3 at approximately 10kgs (unloaded) and the G-3 at a little over 4kgs (again, unloaded and without accessories), these are very similar to your example of the M4 and the M249, except that they fire a larger caliber round and will therefore have much higher felt recoil in both instances. Firing either at full auto from a standing position is highly ineffective if accuracy is your goal.
I would also argue that the M249 is a light machine gun in SR5 terms, not an MMG. It's 7.62mm equivalent, the FN MAG, is more along the lines of a general purpose machine gun, or MMG, at least to my mind.
All in all, I don't think LMGs have a weird place at all, as you have to compare like for like and take caliber of the round into account.
Real world examples:
Comparable 5.56mm chambered LMG and AR: FN Minimi (M249) and AR15/M16/M4 type rifles
Comparable 7.62mm chambered MMG and AR: FN MAG (M240) or MG-3 and G-3 and AK-103 (modernized version of the AK-47, though both of these are chambered for the slightly different 7.62 x 39mm round)
Comparable 12.7mm chambered HMG and "AR": DShKM, GAU-19, or Browning M2 and Barret M82 carbine and M82A1 (or newer variants) rifle
The M134 Minigun could be classed as an MMG based on caliber, but it's high rate of fire, weight, and actual usage as a vehicle mounted weapon places it more in the HMG category.
In SR5 terms, this breaks down into the following based on base DV (note that the MMG and the HMG has superiour AP values by 1 and 2, respectively):
Comparable caliber LMG and AR: Ingram Valiant and Colt M23
Comparable caliber MMG and AR: Stoner-Ares M202 and AK-97 or FN HAR
Comparable caliber HMG and AR: RPK HMG and Ares Alpha and Yamaha Raiden
As evidenced by the above, comparing the Ingram Valiant, the only LMG available in SR5, to an Ares Alpha is unfair because in reality it is a weapon more suitably compared to a smaller caliber assault rifle like the Colt M23.
The same amount of firepower in SR5 as that thrown downrange by an Ares Alpha thus requires a strength of 10 to wield unless it is mounted on a vehicle.