Actually no, it's a little more complex than that:

Regardless of reproductive lifespan, in a population that is in equilibrium (not increasing or decreasing significantly), the average number of (surviving and reproducing) children per individual is going to be 2. So supposing we could live to be 1000 years old, and still reproduce at that age, we'd have an average of 2 generations per 1000 years.

But lifespan and reproductive lifespan are different. Suppose we could live to be 1000, but only reproduce from ages 15 to 65. Then the equilibrium number of years per generation would be right around 40.

Why should genes strive for an equilibrium? Even humans for most of history tried to have as many offspring as possible.

It has nothing to do with genes - every population eventually hits a maximum capacity, after which it normally remains in equilibrium.

I don't think that is generally true. Historically the growth of human populations has been exponential.

I'm curious. What backs up the idea that the average number of children per individual will be 2 regardless of reproductive lifespan?

It seems that I've seen a somewhat(?) common pattern among people who have young families and get divorced that often times have another family (+2 children) when they re-marry.

The definition of equilibrium. Every child has 2 parents. Thus in each reproduction, every parent creates on average half a child (sounds rather morbid). Thus for the population to remain in equilibrium, each person needs to reproduce, on average, twice.