Very interesting to see the lack harmonisation globally! No surprises here.
LAX has several completely unworkable SIDS, GA procedures and S/E procedures (S/E procedures should really be engine out procedures to cater for aircraft bigger than a twin)
Most of the unworkability stems from the floor of a VFR corridor that runs right over the airport, the traffic density at peak times and the tendency to use opposing runways in light wind conditions particularly for freight operations after midnight.
In an emergency, and a critical engine/system failure at V1 is an emergency in my book regardless of weight, the PIC has all the flexibility to do what ever he/she needs to do to maintain safe flight (Legal right).
However as we all know he/she also has the responibility to get it right (Legal responsibility under Duty of care- Negligence Law).
Generally speaking a pilot unfortunate to be in this situation (Emergency at V1) for real had better hope they are in an airspace that has civilised laws and their legal rights are protected otherwise they will find themselves in Jail for a very long time even if they got it right!
My secret to survival after over four decades of safe flying is a very large jar of vaselne that I keep in my flight bag!