The site appears to be back up now.
And yes, at the very basic level, all these settings involve mechs. Lancer would probably be closer in theme to Evangelion, if you wanted to run it that way. Battletech would be a crunchy cousin, due to its roots as a war game.
Yes, original Battletech was simulating large scale mech warfare with collectible minis (let's face it though, most people just liked to paint and have them). The Battletech RPG (to my understanding) is that flavor adapted to a TTRPG.
If you like mechs and video gaming, I encourage you to check out the MechWarrior series. It is the video game adaptation of Battletech.