A comfy chair:
Agreed that while the books say you cannot do it, at the end of the day the PIC is responsible for the safe disposition of the aircraft.
Ok, now you've made the premise vectors for one of the ILS IAPs. The book makes its very clear; you do not turn inbound without a clearance. When being vectored you have a minimums of 3 miles or 1,000 vertical from other traffic.
If the frequency is blocked I would turn final, for the simple point of traffic avoidance.
With a minimum of 3 miles of 1,000 vertical why is a turn onto the localizer necessary for traffic avoidance? You follow the United States AIM (not the Chinese AIM) hold your last assigned heading and get talking ASAP. Also, the controller will soon see you go through final and you will get his attention soon enough even if you cannot get a word in edgewise.
The parallels are likely to be very busy in LAX, and I wouldn't want to go wandering off over there. It is far more likely that, unless I was given the expectation that I would be vectored through the loc, I am going to be given a clearance to turn.
Again, following the AIM is going to keep you from being violated. You turn inbound without a clearance when the place is busy and the TRACON will likely go nuts.
I was based there for most of my career. They really don't like deviations from published procedures at all.