I've read this before.... Honestly, I hate this story. It's like , "hmphff, there are alternate universes... I guess I'll just blow my brains out.". I know it's supposed to be thought provoking... But the only thought it provoked from me was "f@€& this stupid $&@?"
I know , I know- but then, I'm not really known for my literary translations.
Ok, but here's what bugs me.... the world-lines divide for every decision... so why don't you visit ALL of them while you're out, and why don't they keep splitting while you're visiting and why don't large numbers of you try to return after every trip?
(Yes, I know, it's a story and it doesn't work if that happens.... but I'm not a literary genius, and I read this for the first time Mumblemumble years ago and even then I had that reaction)
It is a bit of a logical conundrum, defining that the universe splits with each decision and then having a decision-making traveler leave that universe. Either the traveler is bound to a certain course of action, decision is impossible, or he is splitting his visiting universe.
In any case, you would only get too many travelers returning in a case where one universe is splitting faster than another. Perhaps the homing signal deactivates when the traveler returns to his "home" universe, so once a traveler has arrived it is nlikely another will follow. Of course, instantaneous arrival would cause multiple travelers to arrive in the same universe.
Another possibility is that when a duplicate of a traveler arrives, this counts as a decision and causes the universe to split.
Although I don't buy into this whole "decision" thing. Humans only have the appearance of making decisions. I'm not even that convinced of quantum "decisions". It seems there is likely some underlying system.
The usable fictional multiverses of this type that I'm familiar with all have some concept of elasticity - if you have a decision, but it is of the form "large latte or large cappucino?", your universe does not split - it may split momentarily (for very small values of moment), but then those two universes recombine in some setting-specific way that may or may not be exploitable by further in-universe technobabble. The rest of your world does not care what you drink, and there are no consequences of non-trivial magnitude, therefore when the wave function collapses, you're back to one universe or worldline. You could, I suppose, give the elasticity of the multiverse a rating equal to the magnitude of consequences resultant from any given action that would cause a worldline to spawn.
This only seems interesting if the elasticity is non-trivial; it takes a large change to split a new worldline, larger than even someone ultimately irrelevant living or dying. You have to stop the Sarajevo assassination, or have the Confederacy actually encrypt its battle plans before dropping them on the battlefield, to get yourself a new worldline. Otherwise you run into all the obvious problems and paradoxes you have here.