"Evem 99.999% is not good enough be enough because you will have at least one person that is guaranteed a false arrest and prosecution "
This isn't the right characterisation.
Facial recognition at 95% can be very useful in helping to find people, initiate investigations. It doesn't have to provide grounds for arrest, let alone evidence for a trial.
Of course, there are problems even there, depending on a variety of issues.
Your license plate is being recorded in a variety of places, and for serious crimes, they will definitely use that information to 'look you up' FYI and face-matching isn't that much different.
As for arrest and prosecution, obviously that's another can of worms, but even then, there is a threshold at which we will effectively call it 'evidence'. 99.999% ID + was in the vicinity, plus no alibi, plus motive and prior record? That's a case.
If you think about it - we consider 'video evidence' to be fairly conclusive ... well I doubt humans are actually 99.999% accurate in their ability to matches faces to video! So it adds up.
" When it comes to justice, mistakes are tolerable so long as adequate compensation exists"
There doesn't need to be any compensation if there wasn't any malfeasance or incompetence in the system. To the extent our public servants are all acting above bar, using efficient, fair and lawful methods, then 'compensation' is not really the issue.
"It's better to let actual guilty criminals get away than explicitly and systemically accept even one innocent person being punished incorrectly,"
"accepting any amount of injustice invalidates that legitimacy and authority."
This is not true - by your very own logic. First - a justice system that 'lets people off who are 99.999% chance of being guilty' - implies quite a lot of injustice. Moreover, that in some instances, innocents will go to jail is tragic but absolutely inevitable. Not in 100 years will we have fixed this problem.
Along with some of the racial injustice you mention, you know what is 'unjust'? The amount of crime, particularly murder that goes unpunished in the US due to low case clearances, particularly in violent communities. That means aggressive killers on the block go free, gloat in their crimes, and do it again. Can you imagine having your sibling or child killed, and essentially knowing 'who did it' - but having them go free? This is common.
Facial recognition is not quite ready for prime-time, and it's invasive in ways that license plates are not, so it makes sense to ban it for the time being - however, there are thresholds at which it would make sense. Facial recognition in airports, near high value targets and such systems for alerting police to 'high value, dangerous targets' might make sense. But for everything else it's just not needed.