Maybe this specific episode wasn't as much about Connor, but the baby certainly played it's important role in it. The episode was about Angel, the drastic change in certain aspects of his behavior, but the trigger to said change was first and foremost - Connor. It's true to say that with fatherhood comes responsibility, and said responsibility cannot be underestimated in any sense or form. Like Angel said it best himself - "Someone has to clothe and feed this guy."
And someone has to, and that someone is Angel. The change in him happened so drastically because the arrival of Connor into his life happened just as abruptly. Connor is an enormous part of his life, his meaning to him in one way or another shown on screen many times, and this episode revolved perfectly around Angel's anxiety as a father. A son was something he never thought he could have, he never thought his life could change this way. No, there is no first-time parent who actually knows what to expect, but no case is as severe as Angel's. And Angel does not only want to be the best father he can be, every new parent wants that. He also wants to earn his right to be one, *and* he wants to not repeat the mistakes done by *his* father (whom we all know by now for a fact, Liam *didn't* get along with, in a nutshell).
It's true this episode also centered on his loss of the mission. It is true because he really did lose it. A person often wakes up one day and this day is the first time he gets a complete insight on his life, and it never comes expected. Angel, like many other things along the way, took it to the extremes.
There are very few things in this world, very few people, that are dear to him, that he'd do anything for. If Angel is known for something it is that when a person truly has his devotion, his trust, his love, it's complete, and its pure and it's unconditional. Probably that's one of the reasons its so rare. No one has such strong possession on those things as Connor, his only son, his miracle son. It's a given and well known fact that maybe we didn't always see it being portrayed on screen, but Angel did spend his share of time wondering whether he was worthy of him, of this gift of fatherhood. Fatherhood has many aspects of responsibility, provider is one of them and for whatever reason, it was chosen to focus on. But the message of this episode was much wider, much more general than merely the money shortage. The message touched a much wider range of responsibility from a father to a son.
Was it in a major sense an expression of Angel's devotion towards Connor? Yes.
Again, was it taken to the extremes? Undeniably, yes.
But the end, when Angel, with all of his 50,000$, preferred just lying in bed with Cordelia and Connor and drifting off into sleep instead of discussing finance (which Cordelia, who supported the mission all along, was more than just eager to do), shows that in the end of the day, even he realizes what's important.
Angel: How's my little man?
Cordelia: Oh, he wonders where his father's been.
Angel: Papa's got mouths to feed.
Cordelia: Ah. I have a mouth, too. Always have had one. Kind of known for it and we've fed it fine for years. Yeah. Angel, I'm glad you wanna take care of your son. I am. I just wanna make sure we don't lose sight of the mission.
Angel: Well, we have to earn a living. I mean, not just for Connor, but for all of us.
Cordelia: I agree. But first and foremost we work for the Powers, help the helpless. Promise me we won't neglect that.