The main reason I think it is useful to give some extra in game reason to use lighter mechs, is that without the AI actually using the initiative system effectively, and I see absolutely zero evidence of that, there is no reason to deploy units in expectation of them doing so. I may wish to deploy lighter units myself in order to exploit the potential that the initiative system offers me, and I expect that will in fact be the norm. I expect I will see heavies deployed against me long before I can deploy a lance of heavies. I expect to see assaults deployed against me long before I am able to field assaults myself. I will usually have lighter units than the heaviest I see used against me.
The tactic I will likely use is to use superior fire power to strip away the lighter units from their flanks, to permit me to properly exploit the initiative advantage I have over the heavier units. Then there are any number of tactics to use against them. You being able to maneuver against their position without them being able to do the same is utterly devastating, especially if your lighter units have jump to get behind cover on the next turn before the heavier units can act.
I expect the heaviest lance I will ever deploy will be a lance of three Orions and an LRM boat which 'might' be an assault. The Orions would be the close assault mechs with jump. As 75 tons heavies, Orions out initiative assault mechs. That's three Orions picking their position and fire on one assault that just happens to have a full bar of instability (that LRM boat). The first Orion puts the assault on the ground with an alpha, the second and third pour on called shot alphas. If I make one of those Orions a Grasshopper, things get even more funky. That's one dead assault for zero risk, two at minimal risk (with the Grasshopper).
The same trick of course could be used by mediums (or lights) versus heavies, or lights versus mediums. And that is just the basest of tricks.
The problem is that the AI will never do any of this, so it is perfectly feasible to ignore this level of tactics and just power through with weight and fire power. At least till it isn't anyway, when five skull missions are matching your firepower, your weapon quality, your pilot quality, and your armor quality. If you haven't learned these tricks by then, things may indeed get dicey.
By giving a reward to players who bring lighter units to a battle than they otherwise could, it gives an incentive to players to try it, an incentive without which, they might never even consider. All other things being equal, bigger is always better. Things are not equal in Battletech, but the ways in which they are not equal are very subtle and almost certainly to be missed by most newcomers if they aren't incentivized to learn how to make it work.