Should and could don't mean do. We're not talking about anything logical or truthful or even reasonable. We're talking about Battletech and Mechwarrior. They don't make sense if you're going by what actually should work better.
If you just ignore the truith and facts, you'll be much happier. I like tanks also, but they're meant to be secondary units in btech. It's a design choice, not a physics choice.
Well that was kind of my point, tank inferiority is artificially dictated by rules deigned to hurt them and promote 'Mechs. Sadly 'Mech importance could be plausibly achieved without destroying the abilities of tanks if the designers had just taken a little time to think about it.
Thanks to their articulated legs 'Mechs have built in shock absorbers, their humanoid design means they can be controlled in freefall the way a human can and non-humanoid designs and humanoid designs alike can mount thrusters to manipulate the fall. The upright design allows for rocket style thrust assemblies to help slow decent, and of course you always have parachutes. With 'Mechs being upright you can more quickly fill the same volume of transport space by removing several decks and the elevator assemblies vehicles would need. The fewer crew members (1 instead of 3 minimum) means you have fewer people who need to be briefed about mission details helping with OpSec.
All of these points make the 'Mech perfect for: securing beach heads/LZs, conducting hit and run attacks on other worlds, rapid redeployment to regional hotspots on a world and advanced operations behind enemy lines.
The strategic potential of the BattleMech is enough to ensure its importance. In the skirmishes and border wars that followed the Second Succession War these traits would become even more important as full scale deployments and conquests would be harder and harder to achieve. Spoiling attacks and raids to keep the enemy off balance or to secure resources would be more important and missions for which the BattleMech is better suited to.
You also have the psychological effect of BattleMechs. Giant towering machines of death and war, that are just freaking awesome and they're piloted by one person. They will resonate more with civilians who can more easily imagine themselves being a hero in one of these things rather than being one man on a team in a tank. Add in the fact that 'Mechs see more use thanks to raids and being the "tip of the spear" for invasions, their mystique will grow to point of hero worship.
In game play terms you just have to tell the players: "you want to drive a tank? Ok get your friends together, you need one to be a driver, he'll pick your route, you need one to be the gunner, he'll pick your target, and you can tell them what the mission is and look for new threats." Everyone will play 'Mechs if you do that.