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I actually strongly disagree that "it's always worthwhile to pick up a random card at no cost". If it has no useful purpose except as a potential hitpoint, then if I'm not in danger of dying, I'd far rather not dilute the effectiveness of my deck by including it. Plus hopefully its basic and I can banish it and never see it again...

I have some specific examples I'm curious about, using everyone's favorite monster, the Elemental Arachnid (You may play any number of armors during this encounter. Before you act, you are dealt 1 Acid damage, then 1 Cold damage, then 1 Electricity damage, then 1 Fire damage).

1. If I have (only) Reed Snake Armor in hand (reveal to reduce Acid, Cold, Electricity or Fire damage by 3), do I take a total of 3 damage or 0 damage?
2. If I have an Elemental Treaty at the location (While displayed, reduce Acid, Cold, Electricity, Fire and Poison damage dealt to characters at this location by 1) and no armors, do I take 3 damage or 0 damage?
3. If there are two Elemental Treaty spells at the location cast by different characters, do I take 3, 2, or 0 damage?
4. If I am at the Stonework Passages (Damage dealt to you is reduced by 1) with no spells/armors, do I take 3 or 0 damage?
5. If I am playing a scenario using Defensive Stance (When you are dealt damage, if the damage is not reduced to 0, banish the top card of your location deck and each unoccupied location deck) and I have no armors/spells to reduce damage, do I lose 4 boons per location or just 1?

There is a Chimera villain in WotR adventure 2 with 3 checks as well.
But I don't read it as you only being able to use one other character. It says other checks may each be attempted by any one character at the same location. I think its just emphasizing that you can't have multiple characters attempt the same check.

I think its no different to the Regatta being unwinnable in S&S if you close a location. Its not like this can happen to you accidentally. If you close all the other locations before you go to the Chasm, well, that's your fault.