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Maybe it's a matter of perception. I understood his reasoning to be a bit of both, i.e. you don't want anybody to know you're a hitman, *especially* your children.
As for the son participating later, that's precisely the reason why he's in the moral crossroad later on. He's tried so hard to protect his children from this "perdition" by keeping them at a distance, but the events "force" him to push his son towards the same thing he had been protecting him from. At this point, he has no choice, it's either let him participate and survive, or not and they'll die. But that's precisely why he's morally afflicted and why the events in the last act unfold the way they do. He's trying to drag his son away from that.
As for the son participating later, that's precisely the reason why he's in the moral crossroad later on. He's tried so hard to protect his children from this "perdition" by keeping them at a distance, but the events "force" him to push his son towards the same thing he had been protecting him from. At this point, he has no choice, it's either let him participate and survive, or not and they'll die. But that's precisely why he's morally afflicted and why the events in the last act unfold the way they do. He's trying to drag his son away from that.
If there were conversations involving him explaining that was the reason he kept his criminal activities secret from his kids, it could be argued that's part of the reason, but in the end, I'm left with what the film provides to judge what his motives are and, based on what we see, it seems like his reasoning revolves around fearing the threat this knowledge could cause to his family if his boss were to find out.
As for him bringing his children along, while that does feature him indirectly involving his son in his criminal behavior, this would only add to this theme of him fearing that his kids would grow to be like him if this theme took priority in the film beforehand. I didn't get the same connection you did though for reasons stated above, so I didn't have the same reaction to these scenes you did. It wasn't until his conversation in the church basement that this theme took priority for me.
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