From the backstory we were given of Homelander during “The Boys,” we learn that he was raised in a lab, and he was always under the eye of scientists. At no point was he given the nurturing environment that he needed while growing up, and something like this is bound to have an effect on anyone who has had to experience it.
It is most likely that it was because of this environment that he turned out the way that he did. Not only does it color his relationship with Madelyn (and the way that he has seemed to latch onto her as a nurturing, mother figure), but it also colors the way that he feels about other people. Except for Madelyn and Stormfront, he does not seem to care about anyone else in the whole world; the two of them are the only two in the world that give him any kind of affection (whether real or feigned). Both of them manipulate him to get what they want done, and that seems to be the only way to get him to do anything other than what he wants to do.
Could things have been different for him? If he had been given any kind of affection when he was younger, it seems likely that he would not have ended up being the horrible person that he became; he would probably not have become a complete sociopath who enjoys hurting other people (and who is willing to kill loads of people to do what he thinks is the right thing, in his own warped sort of way).
With the hint that his son (the one that he had with Butcher’s wife, Becca) might have the same powers that Homelander has and having at least one parent around that actually loved him, we might get to see what would have happened to Homelander if he had been in the same situation. But if Homelander finds wherever Butcher hid the boy, he might be in danger of Homelander doing something horrible to him (not only because he is a sociopath, but also to relieve whatever frustrations he had from the deaths of Becca and Stormfront, and Butcher spiriting the boy away from him).
What might we expect from Homelander in season three? He will likely continue to harm other people with no regard for anyone else, but what might happen if he finds where his son his hidden? Will he try to kill his son in his anger about what happened with Stormfront, or will he continue to try to turn the boy into a smaller version of himself? Both scenarios seem plausible, and it would be preferable for him never to find his son (and for the boy to come back and stop Homelander from harming anyone else).
Without someone being introduced who can stand against him, he is probably going to remain a danger to anyone who comes into contact with him.
In the comics, it is Black Noir who is the only one who has the power to put a stop to anything that Homelander might do, but Eric Kripke (the showrunner) has stated that the writers are planning on going in a different direction with Black Noir. So, we are going to have to look for someone else who might stand up to him.