> Please explain what prevents Mexico from improving its standard of living on its own.
Nothing whatsoever. Except of course that no country exists 'on its own'.
> Mexico is not some kind of slave to America; it's a sovereign nation, in control of its own destiny.
No, it's a lot less in control of its own destiny than America, mostly because of America (incidentally, also the biggest consumer in the drug trade). To all intents and purposes Mexico is utterly dependent on what happens on the other side of the border. Of course it is easy to deny that if you choose to do so but it suits quite a few interests in the United States just fine to have a weak and poor country next door.
To the point that there have been serious interventions in Mexican internal affairs to keep it that way benefiting (mostly American) foreign strategic and economic interests.
> It's free to trade with all the other 200+ nations on Earth, or to attempt to negotiate having its people go live there.
To the extent that it is able to do so, it does.
> Why does America get all the blame in your world for Mexico's problems?
It doesn't. Why do you read it as such?
America is to blame only to such an extent that they first establish a whole pile of factories abroad when it suits them and then at the drop of a hat imperils the newly created stability and middle-class in a country next door. That sort of action comes with responsibility.
> Who does America get to blame for its problems?
With great power comes great responsibility. If you don't want to be seen as responsible do not continuously mess in other countries affairs and when you do realize that you can't withdraw that support overnight.