Confusion as to what the sentence means

I have enough trouble overcoming my own limitations.

That's enough for me. So I make no effort to ask why God didn't distribute intelligence evenly among men.

I have enough trouble doing X without Y.

This kind of frame is used to mean the following. X is already bad, Y would or will make the situation worse. The illocutionary force of this is usually ...

• X is already bad, why did you do Y and make it worse!

... or alternatively ...

• X is already bad, I am not going to do Y, which would be even worse!

For example:

• I'm busy enough without you coming in and asking me all these questions. (Why are you being such a pain in the neck?)

This would be an example of the first. An example of the second would be:

• I'm late enough without stopping off at the store. (Therefore I'm not going to stop off at the store).

The Original Poster's example is of the second type:

I have enough trouble overcoming my own limitations without fretting over the fact that God has not seen fit to distribute evenly the gift of intelligence. (Therefore I'm not going to fret about it).

This could be paraphrased as: I have my own problems, I'm not going to make things more difficult by worrying about why people don't all have the same intelligence.

How to Win Friends etc is a classic that sold millions of copies. Your basic analysis is correct. But Dale Carnegie is making some other points:

(1) if you are smarter than other people that is no credit to you;

(2) people do like it when you tell them they are stupid

It is a variant on the mote and the beam (Mt 7 v1-5.

It means “I have my own shit to worry about, I’m not gonna waste my time worrying that not everyone was given the gift of intelligence.” Basically saying “It doesn’t keep me up at night that you’re stupid and I’m not, ive got my own shit to deal with”