So, in the Cataclysm scenario, a LOSS condition is if any one player is eliminated. However, in the Brother against Brother Event card, it says you 'may' attack another player and if he eliminates them, he immediately wins and all other players lose.

Does this card basically overrule the base conditions of the scenario?

I'm assuming this is what the description means by: "Just be mindful of selfish, backstabbing players who are only looking out for themselves."

Yes, it overrules the base condition of the scenario. So if one player is at 19 damage, and another player is at 5, the player at 5 may wish to just destroy the 19-damage-player to ensure a win for himself, since that 19-damage player may end up dying, causing everyone to lose in the next round.

For a truly cooperative game, you should remove that card before play. For me, I like to have the contingency there so I can choose to take out another player if I wish. But in a lot of cases, you run the risk of attacking them, not defeating them, and have weakened both of your hands for the upcoming events. So if it's not a sure-kill, it's really not in your best interest to attack someone else. Also, if it does come up, and nobody decides to attack anyone else, you basically got a short breather round before finishing off the scenario.

So if one player is at 19 damage, and another player is at 5, the player at 5 may wish to just destroy the 19-damage-player to ensure a win for himself, since that 19-damage player may end up dying, causing everyone to lose in the next round.

You cannot attack players having more damage than you, if I recall correctly...

This is exactly to prevent the "pray on the weak" tactic you described above. If you want your "betrayal win", you'll have to fight hard for it

Woops! You're absolutely right! I guess I really do ignore this card more than I thought. I play solo mostly so it kinda makes sense. Hmmm. With that extra restriction, I wonder how often people win by Brother Against Brother.