Reason alone is sufficient, but research and investigation haven’t turned up any other way for this to work.
We have two points where Jesus may have saved humanity in some way: his death and his ressurection. We’ll take these one at a time.
So. What is the function of Jesus functioning and/or dying? What does it cause, and what is the mechanism by which this comes about? Well, there’s only one mechanism that a sacrifice functions through - by pleasing the god it’s a sacrifice to. There are two mechanisms this can function through; pleasing the god directly by the sacrifice itself, or pleasing him that his servants are sufficiently devoted as to suffer loss just on his say-so.
In the first case, God ‘eats’ the sacrifice, enjoys the flavor and taste and smell, and is made happy. If this is the argument, then God is bloodthirsty - kind of scary. Even put as charitably as possible, blood and pain become a currency that God trades in; you pay in blood and buy a reward. In theory this could work - though it’s a sick, sick god that trades in pain. If we choke this down, though, we can continue: skip down to “10”, below.
In the second case, the people killing Jesus weren’t showing devotion to God in doing so - Jesus even feared that God would be wrathful towards them for killing him, asking for their forgiveness. This leaves only the possibility that Jesus himself was the one making the sacrifice (which can be reasonably argued), and with his own death was earning the favor. This works reasonably well: goto 10.
10: We now have a god that’s been pleased by the sacrifice of Jesus, and then proceeds to forgive humanity. But wait - this means that God could have chosen to forgive humanity anytime. The sacrifice of Jesus didn’t enable the event; it just prodded God to do what a good God would have done right off the bat. This is consistent with the jealous and angry bronze age tribal diety that God originally was, but not with any benevolent diety such as the one he’s been whitewashed and painted as being now. And even if we choke that down, we’re still left with there being nothing particularly special about the Jesus sacrifice; nothing about the sacrifice itself changed humanity’s state.
Looking at the ressurection helps even less - I’ve heard no explanations of how it was supposed to cause salvation. I’ve heard it’s a symbol of salvation, or a proof of or demonstration that Jesus had the ability to transcend death - but then even if he hadn’t done it, he still had the ability to transcend death, or he couldn’t have ressurected himself. And there is of course no causal connection between Jesus rising from the dead and the rest of humanity’s relationship with God changing; that just doesn’t make sense. I honestly see nothing here to make an argument based on; somewhat disturbingly the best argument I see for it here is kanicbird’s that once he said he would do it, he had to do it, or else become a liar. (Though I’m not sure how it matters if he’s a liar.) Regardless, in that case he only had to be reborn because he shot his mouth off beforehand; kind of a flimsy pillar to balance a salvation on.
Probably the best non-classic argument I’ve heard justifying Jesus’s claimed role as a salvational figure is that only by living on earth and suffering and dying did he acquire the sympathy to be able to argue our case before God as an intercessor. The thing about this scenario, though, is that it presumes that God doesn’t understand our pain, which is a problem for both the sparrow-falling diety, not to mention omniscience, which is a lot to throw out of the God definition just to justify the Jesus thing.