Proof of physical presence in the U.S.A. not accepted

Also, the only thing that worries me by going to Mexico City and get my passport (for sure I will get it there), is that my kids and wife are living with me in the USA with my working visa (addendum) and as soon I get my passport that same day, my and their visas will be voided. What is going to happen to my family then? I need them to be here with me and going to school (my kids) and my wife with me at home. What would be your recommendation then?

When you return to the US, you file an I-130 and I-485 for all of them, assuming your children do not qualify for US citizenship.
 
Another option is to apply for N-600. Some people who failed to directly get a passport had success with the N-600. It takes a longer time and costs much more, but apparently those who process the N-600 are more familiar with the ins and outs of acquired citizenship and take more time to review the facts and the law.

Note that if you are older than all of your siblings, it is entirely possible for all of them to qualify for it based on the same evidence although you don't qualify, without there being any contradictions. The older law required 10 years of presence, while the newer law requires 5. In addition, the years your father spent in the US after your birth could count towards eligibility for your younger siblings, but not for you.

Whatever route you choose, do what you can to gather more evidence.
 
You are so focused on getting them to accept those pieces of evidence you already have ... is it impossible to get other evidence, like employment or school records?
 
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