Is there a merit selecting an Applied Physics/ Engineering Physics major over a mechanical or aerospace engineering major career wise in the future? Which offers more critical problem-solving requirements for these jobs?

Excellent question Kai. With a little more research, I think you will find that each one of those will deal with a slightly different area of interest that may or may not appeal to you.
Aerospace engineering is obviously going to be mainly dealing with some sort of aircraft, or aircraft related items. This is all great, until you realize that there are only a hand full of manufacturers world wide and only several other research centers that may be looking for an Aerospace engineer at the time that you are job hunting. Now, if you are like me, and eat, breathe and sleep aircraft, and could not think of anything else you would rather do, that this would be the obvious choice. From personal experience, don't get involved with aviation, unless that is what you really want.
Mechanical engineering is a field that can be applied anywhere. If you look around you, pretty much everything has had some mechanical engineering applied to its inception at some point. In my opinion, this is the widest field with the most opportunities available out there.
I don't have a lot of knowledge about the physics engineering, but I believe that has to do with a lot of abstract ideas, formulas and things that you may never be able to put a hand on or see a finished product.
You will have to make a decision on which seems like the most appealing one to you. All engineering will deal with problems, but some might interest you more than others. If you really like cars, than aerospace might not be a good fit. If you really like astrophysics and are always thinking of how to get to Mars faster, than mechanical engineering may not be the best option.
Good luck!

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