The main downside I can see from your suggestion is some players having difficulty determining if they're actually benefiting from Basic Instinct.

Yes, I know they could simply move or verify by comparing values of heals/damage numbers with BI on or off (or even Mighty Guard on or off), but that may be the reason that it functions the way it does.

A net benefit given to players with this change would be that having the last BLU alive in a group would immediately benefit from BI (instead of having to hang onto the spell the whole fight and cast it after everyone else dies, Stinging or otherwise) and that may have been deemed too powerful. I don't think they balance BLU content to the point that they're worried about giving players access to "9 Final Stings instead of 8", but there's at least some attempt there given they refuse to let BLU to queue into Ultimates, Deep Dungeon or even Memoria Miseria Extreme (presumably, Eureka and Bozja would potentially interfere with players that don't want to play with BLU players which they couldn't avoid where the other things would all be pre-made groups for BLU to enter).

Besides all that, I don't think there's any buffs in the game that work like you're describing, though: ones that only actually apply in the correct circumstance but don't if you're not in it. So, they might not be able to simply make it so. I don't even think there's any retroactively checking buffs, if they wanted to maybe split BI into a BI (dormant) and BI (active) buff that checks when you're in a proper scenario / your teammates are all downed. I may be wrong, though.

As far as the speedrunning issue, that sounds like a bug and I suggest creating a bug report regarding it. I don't understand from that explanation myself, considering it seems impossible to queue and enter a new dungeon before the buff falls off upon exiting a duty, but if you can recreate it consistently then I'm sure that would be addressed. Not that it sounds like that's particularly hurting anyone and folks that enjoy it may be sad to see it go, so I'd understand if you didn't want to write it up.