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.
If the frequency is blocked I would turn final, for the simple point of traffic avoidance.
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.
If it was Singapore, where you frequently get vectored through the localizer, then its a different story.
Very much situation dependant.