Interesting that you mention your volunteer diving at the NY aquarium. Our local aquarium, in Long Beach, has a volunteer diver program but to qualify we have to be at least Rescue certified. This requirement seems to make sense, in light of your questions... because I agree, it would be reasonable for the aquarium divers to have some basic rescue diver training. Our DSO thinks so also.
PADI AOW requires five dives. Deep and Navigation are required; the remaining three are chosen from a list of about 20 options. If you only did three dives and were then given your AOW card... bad for you and them.
The AOW class is not designed to make you an "advanced" diver; it is designed to simply advance your experience with different kinds of diving. And to be brutally honest, the value of the AOW class really depends on how much effort the instructor wants to put into it. All too often an instructor will do the bare minimum, just to meet the minimum required standards, and the students will be satisfied because they get a new card. But if the instructor wants to put out some effort, their students can really feel like the AOW class was worthwhile.
With that said... if you want to learn Rescue, take the Rescue class. But find the right instructor. If you have an instructor that only wants to give you the bare minimum, you might feel like you're missing out.
This should have been covered in Open Water class. If you mean how to ascend at a proper rate without looking at ANY instrumentation or any reference (i.e. just open water)... if anyone tells you that is taught anywhere, they're having one on you.
The reason these things are not included in an AOW class, or even a Rescue class, is that very few people are interested and even fewer people have the need to learn these things. I've never ascended feet first, never had to breathe off a hose or a cylinder. These are not things that are generally useful... so why would PADI include them as standard requirements in an AOW class?
However, breathing from a freeflowing regulator should have been covered in your OW class.
You might also find that no instructor will do the latter two (breathing from a bare hose or tank) with you because they are not a standard part of a course and if something were to go wrong while you were doing this, the instructor would have a LOT of explaining to do.
Open water class. Predive equipment check should cover things like this, preferably so you find the problems before you get in the water, and cover how to fix simple problems.
Loss of viz and/or mask should have been covered in OW class... part of dive planning and the buddy system. In fact, a "no mask swim" should have been one of the skills you performed in confined water during your OW class. If you completely, suddenly lose all visibility... thumb the dive, make sure your buddy knows, and ascend. This is simple OW stuff.
Night dive is one of the options for AOW, but it is not required. Your instructor should have gone through the options and allowed you to choose a night dive, if conditions permitted. Of course you can always take a night dive course separately.
Deal with getting stuck should have been covered in OW class. You shouldn't be getting stuck in a wreck because you shouldn't enter a wreck without proper training. THAT should have also been covered in OW class.
Open water class?
As I've outlined above, the necessary things on your list should have been covered in OW class. The others are not generally necessary. If you find regular occasion that requires you to breathe off a cylinder or hose, something has gone horribly wrong with your dive. Which means your dive planning (something else covered in OW class) needs improvement.
The Rescue class is designed to teach self-rescue and rescue of others. That is the purpose of the class. The AOW class is not designed for this. Suggesting that the Rescue course is not worthwhile because AOW did not teach any rescue skills... doesn't really make sense. If you want to learn rescue skills, take the Rescue course.
How do you avoid panicking? Training and experience AND dive planning. Always make a plan with your buddy, follow the plan... just as you learned in OW class.
As for classes being more rigorous: there's a lot of debate about that. The upside to classes now is that we know a lot more about diving than we did a long time ago. So one could argue that we're a lot better at knowing what is essential for certain courses and what is non-essential (i.e. what topics and skills are better left for a later course.) Most people that offer an opinion of the difference between dive training now and then are just making noise. To make a truly objective comparison, they would have had to take a lot of training, from many different instructors, a long time ago; and then take a lot of training recently, also from a lot of different instructors. I don't know anyone that has done this.