To install a new chain I need a chain tool to shorten it to the proper length. Why not just use that same tool to replace the pin and not deal with a magic link? Seems to me the pin would be stronger than a breakable link. Am I missing something?
You can do that (*) - you just need to make sure you don't push the pin out all the way before reinstalling it (else it becomes quite hard to put the pin back in). However, a quick link is easier to install and remove (**) the chain (also, carrying a few of them around make for easy emergency repairs), since you only have to use the chain tool to shorten the chain and not risk damage to the plates when re-attaching or not aligning the pin correctly when re-attaching (this is the most common cause of failure for a chain, especially on higher speed chains). Thus, re-installing the pin is more likely to weaken the chain vs a quick link (which doesn't have this problem).
In any case, the quick link isn't usually the place of failure, so theres not really a reason not to use it unless your chain doesn't come with one.
(*) Well, check the documentation that comes with the chain. Normally, if there is a quick link, the manufacturer will tell you to use it, which the end of the paragraph will tell you why.
(**) On 10/11 speed systems, they are normally one time use links. On lower speeds, you can open and close the chain a bunch of times.
(***) As the comments point out, Shimano chains ship with a special pin to use (as do some Campy chains). You have to use this pin in this case (see this link for details/intricacies).