[MortMan]

The overhead bin space would be taken up.

The airlines do not want to fight passengers to put stuff where it should be put - most have rules against their flight attendants doing that. If the coach class passengers are wandering through the first class cabin and the bins are empty, while the coach bins are mostly closed - then they will stow their luggage in first class.

[I_Publius]

Anything over 3 hours and I seriously consider 1st class. Especially on AA.

[Blue Jays]

That is easily resolved.
If baggage cannot fit in the class where you purchased your ticket, under the aircraft in the cargo hold it goes.
Easy-peasy.