If you want to use light mechs and make them useful on the tabletop, here's what you do.
Get together with your opponent and agree to scenarios that make logical sense for light mechs.
Somebody earlier mentioned the damaged Thunderbolt headed back to base, that encounters a lance of 20 tonners. That's a perfect example of a fun battle that will be relatively balanced. You don't have to play out a double blind scenario where everybody is trying to outmaneuver the other. Simply declare that the battles you will play out will be a portion of what is going on in the invasion. Let the lights get a moment in the sun, and then assume that's what they're doing the rest of the war.
Take a Stinger and a Wasp and throw them against a company of infantry and a Scorpion tank. Your justification is that it's a raid on a small munitions outpost. The armies are fighting with what they have, and while some super-hovercraft would probably be really good in that situation, all the super-hovercrafts are currently busy. Scarcity is a real thing in war.
Use an objective in a fight. The attackers are coming through a ravine and they have to get past defenders to get to the other side. Put 4 mapsheets end to end long-wise, with the light mechs coming in from the side of mapsheet 1, and the defending mechs starting on the line between mapsheets 2 and 3. The defenders have more tonnage, but the attackers don't have to destroy any defenders. The attackers have to get off the edge of mapsheet 4. Once past, they're considered to be able to outrun the defenders and they'll have a straight line to the target (so they win). So it's just a battle of maneuver for the attackers, while the defenders have to destroy them.
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As far as vehicles, light mechs have the advantage that they resist attrition better. A hovercraft or a VTOL can move faster and carry more weapons, ton for ton. But let's presume that we're going to use our mechs to fight intelligently. Not fun, but smart. Something fast with an AC 2 would be an absolutely lethal machine, as long as you had hours and hours and hours to play out every single engagement.
Take a stock 3025 Vulcan battlemech. Generally one of those mechs that nobody wants to take. But it has 6/9/6 movement, okay armor, and an AC 2. What is his job? His job is to snipe and harass vehicles from very long range. Take a shot at every VTOL that flies by. A lance of hovercraft? Shoot at 'em. A column of heavy tanks rolling down the street? Empty your ammo bin shooting at them. Basically you fire until the gun is empty and then you hightail it back to base. What good does this do? You're going to cause motive hits, the occasional critical, knock out a few weapons, things like that.
On its own, this one mech isn't going to do a whole lot. But in an environment where there's always some jackass with a little gun plinking away at you, those things start to add up. A battlemech will very rarely take any sort of serious damage from these harassing engagements (there's always the chance for a through-armor-critical, but not nearly like with a vehicle). So the battlemech can almost always return to its dropship and have armor replaced. Top off its ammo bin, and the mech is ready to go out again. But with vehicles, you get that hovercraft that got immobilized 200 miles from the city. You have to send people out to recover it. You get VTOLs that crash. Turrets get stuck and weapons get destroyed.
Low intensity engagements heavily favor battlemechs over the long run. While we like to play the "big smashy robots fight to the death" type games, mechs are best suited to deny those types of fights to vehicles. While nobody wants to play that out (who really wants to play a Vulcan shooting at 24 hexes, needing 11s and 12s to hit a group of Rommel tanks for 45 turns, hoping for a lucky crit? Nobody, that's who), we can presume that those sorts of boring battles are happening offscreen.
Why do I bring this up? That's your justification for why people don't flood the battlefield with tricked out vehicles. Light artillery fire, some prick with a snipey gun, somebody who lays down some Thunder LRM-5s scattered across a battlefield, the 31st century's equivalent of a D&D "wandering monster chart" whittles down vehicles. It's this environment where light mechs really thrive, because they're less vulnerable to a random 2 point or 5 point damage location.
