There is probably grain running somewhat side to side vs. running completely end to end. If you were to cut a new slat that was parallel to the wood grain, it would be much stronger. And of course, if you used a stronger wood, that would help, too.
[...snip...]And then...
[...snip...]Elmer's yellow woodworking glue vs. Titebond? Pretty comparable, really. Elmer's should be "stronger than the wood", same as Titebond. See below for some supporting data.
I am guessing Lew suggested epoxy because it is strong even when you have gaps, good when repairing cracked wood such as you have.
For some general info on strength of glues, here's a link to a test done by Fine Woodworking (the link works only if you pay for their web extras).
Titebond III SystemThree T80 Epoxy Elmer's Carpenter's Glue Liquid "Old Brown" Hide Glue. Moser's Hide Glue Gorilla Glue
on Maple, Oak, and Ipe. For each wood, they tested on tight, snug, and loose joints.
The Titebond III and epoxy were neck and neck througout the test, with Elmer's a close third.
In general, all the glues were within shouting distance, although Gorilla Glue had poor results on loose joints and was overall the weakest.