That would help, but a lot of people have such a distaste for checking luggage, even when free (And TBF to them, it can easily add an hour to your trip), that I don't know it would completely solve the problem.
> TBF to them, it can easily add an hour to your trip
I used to care when I was in my early-mid 20s because I'd often take evening flights in the day before I had to work (PTO was at a premium). Having to wait an additional 30 minutes to an hour for bags sucked when it was a two hour drive home and it was already 10:30. Now that I live next to the airport and have more leeway as regards my leave I don't sweat it so much.
I fly Southwest specifically because they don't charge to check bags (seriously, fuck this nonsense of charging for basic functionality). I find that it's actually very rare that the overhead bin space fills up, even on completely full flights. I think that the GP is spot on that contention for the bins wouldn't be nearly as bad if checking bags was free (which again, it bloody well should be even if it didn't help the bin situation).