Croesus
Adventurer
I'd check carrying capacity. If you're not overly encumbered by the addition of a bard's worth of new weight, hunky-dory. If you are, spell is suppressed and you both fall unless you can kick off the bard. That way, it's better to jump on the feather falling stone giant than the 6 strength halfling. Interestingly, this adds a fun tactic for suicidal creatures to use against feather falling enemies. Just pile on until they cause the spell to fail.
But, yeah, I agree entirely with both your requirements of a check to pull it off and the concern about action economy. As presented, I'd not let the Bard attempt it due to lack of actions. Next round, sure. And I'd call for an opposed check if the feather falling villain didn't particularly want a passenger.
Whereas I'd approach it from a slightly different perspective.
FEATHER FALL
1st-level transmutation
Casting Time: 1 reaction, which you take when you or a
creature within 60 feet of you falls
Range: 60 feet
Components: V, M (a small feather or piece of down)
Duration: 1 minute
Choose up to five falling creatures within range. A falling creature's rate of descent slows to 60 feet per round until the spell ends. If the creature lands before the spell ends, it takes no falling damage and can land on its feet, and the spell ends for that creature.
What feather fall does is change how gravity affects creatures. An affected creature descends slowly. An unaffected creature descends rapidly, unless it can grab onto something - a railing, a ledge, a rope, or a feather falling creature. Strength matters for who is trying to fight gravity. If the FF creature is holding on, either they succeed, or they lose their grip and whatever/whoever falls. If the grappler is trying to hang on, either they succeed or they fall. In both situations, all we need to know is who is/is not affected by gravity and who is/is not attempting to hang on.
Of course, this could lead to some silly results. Fay Wray falls from the Empire State building, but casts feather fall. King Kong jumps after her, grabbing on. Do they plummet, or do both go crashing to the ground? Hey, it's magic. And feather fall doesn't say "up to five falling Medium creatures", it says "up to five falling creatures".
Besides, it would be kind of cool to see King Kong floating down holding onto Fay Wray (or Naomi Watts, take your pick).
