Ideally, all airline flying should be done by that airline. A good example is Lufthansa. All flying, whether it is a flight on a CRJ900 or overseas international flight on a B777, are flown by Lufthansa pilots under one seniority system. If we had the same system here in the US, there would be no Mesa, GoJet, XJT, CHQ, Skywest etc, etc. All United Airlines flying would be done by United Airlines pilots and all Delta Airlines flying would be carried out by DAL pilots whether it is Saab 340, CRJ900, CRJ200, or a E175 flight. This way all training would be under the mainline control and everyone would go through the same type of training.
Everyone gets a random selection on what type of aircraft to fly, giving consideration to their own experience and desires and based on that particular airlines' needs at that particular time. Similar to the aircraft assignment in the military. If you do well during the initial ground school before the aircraft systems training begins, you get your pick of the aircraft, within the constraint of the airline needs at the time you finish your initial ground school.
On the point requiring ATP to fly Part 121. Yeah, I agree with that. That takes care of minimum 1500 hours and min age 23 requirements.