Sorry for replying so late, but I hadn't been in the mood to check boards lately.
Basically, a light 'Mech has mobility on its side. It can get around to the weaker rear armour of the heavier 'Mech at will and it can choose the engagement range. That's the chief advantage of the Light 'Mech.
Remember that the Classic BTech is a mathematical chessboard and it becomes equations. In more practical terms, a Light 'Mech with say, 8/12/0 (Locust class) would be able to walk around a Heavy.. say, Warhammer and pot at its back.
And you say that the 'Hammer would be able to torso-twist and bring at least one PPC to play in reaction? Yes, that's true.. but assume a straight 8-hex movement, that's a +3 modifier there straight away, added on top of the 'Hammer's own movement modifiers. If the Locust is under the minimum range of the PPC, there's additional modifiers.
Assuming a realistic scenario where the Locust walked 6 hexes (it had to turn), and is up right next to the rear of the walking 'Hammer, the 'Hammer can only bring one PPC to bear at +6.. so that's 10s to hit for an average IS pilot (BS4). In return, the Locust gets a +2 modifier, so 6s to hit. Much better and on the rear armour too.
This is talking about 3025 tech, which had more disadvantages to the Light 'Mechs. Bring in Clantech with its pulse lasers and targetting computer, the Light 'Mechs are much better. Weight saving technologies benefits Light 'Mechs better than Heavies, to the point where Light 'Mechs moving at 9/14/0 (or more!) with enough firepower to core through the back armour of even an Assault 'Mech in a salvo or two.
However, a good Light 'Mech pilot needs to know when to press and when to retreat. As you noted, Light 'Mech does not have the armour to take punishment, and the normal mistake that new players do is to stick in a firefight which they can't pay for the exchange rates.
Heavy 'Mechs have more room for errors and is generally more fun to play for the casual player. For more competitive players, Light 'Mechs requires more finesse and skill.