For the degree that the television series made Tywin Lannister a very intelligent man, he was an idiot in this most basic regard: he didn't remarry after his wife died after Tyrion's birth.
According to the series Wikis, Tywin is age 67 in Season 4. Tyrion is age 39 in Season 8. Let's assume that in Season 4, that Tyrion was 35 years old.
That means that Tywin Lannister was 32 years old when his wife died.
This is just stupid. What man in the fantasy world at age 32 would choose not to get married and have more children, especially considering his wealth?
Cersei and Jamie are four years older than Tyrion.
This is in contrast to Walder Frey (https://awoiaf.westeros.org/index.php/Walder_Frey):
Walder has had twenty-two trueborn sons and seven trueborn daughters from his marriages, with an unknown number of bastard sons and daughters.A lot of the drama in the series seems to revolve around families that have a small number of children.
If I were a King, or Lord of my own vassal, in that kind of world, I would sire as many children as possible.
The stakes were so much higher when the only son dies, versus if there is a football team ready to take the field.
This was how things were done in Old Testament times and makes the most logical sense.
Tywin should had read Psalm 127:3-5 and taken it to heart:
Behold, children are a heritage from the Lord,
the fruit of the womb a reward.
Like arrows in the hand of a warrior
are the children[a] of one's youth.
Blessed is the man
who fills his quiver with them!
He shall not be put to shame
when he speaks with his enemies in the gate.