So I go check the Military Base on the North West region of the map, find the Military Base full of mutants and return to the Brotherhood of Steel Elders to inform them, they tell me that I should talk to Maxson (the old guy in the 4th Floor) to give me more instruction but the script that triggers that quest doesn't seem to work. What can I do?

Talking to Maxon isn't required after the conference with the elders. Just head to the base and you'll have your support, and even if you don't have them the don't do much. Capping the Muties that are topside is easy to do if you snipe them in sneak mode through the fence.

Depending on your character you can confront them all at once if you have hardened power armor and a turbo plasma rifle.