Yes it isn´t really intuitive that mages can beat archers, but you have to look at the deatails and it makes a lot of sense. I often use Mages to beat enemy LR
1. Mages are bigger units than LR, so 10k Mages are actually as strong as 30k LR troops.
2. It depends a lot on the type of LR troops we are talking about: regular elven barracks LR troops have 0 defense bonus against mages. So if you use your barracks LR to attack mages you have to make sure your strike first. (You have a 90% attack bonus). basically in this fight whoever strikes first wins. In general that should be your archers (higher initiative), but terrain obstacles can prevent you from hitting the enemy. In that case you may walk right into the line of fire of the mages and they´ll shred the LR troops.
If you have Rangers, you get a LR troop that is actually designed to eat mages for breakfast, they have both attack and defense bonus, they have double as strong defense down debuff for enemies and on top of that they have a bigger walking range.
So if you have trouble either use manual fight on the computer or use Rangers. Or hope for a better geography next time
Since you mention also LM troops: They suck at everything
So yes they will also lose against mages, they lose against everybody. Dogs can stand up to mages, but nothing else, Drone Riders do OK (but not more than that) against LR troops and the barracks LM troops lose against absolutely everybody...
But all of that is not only bad, you can use it to your advantage, supercharged Blossom mages (or to a lesser degree barracks mages) can rip through basically everything you encounter (although you still want to avoide running into Mist Walkers and dogs).