Me personally, I would install backing on one side, unless you have some compelling reason for needed access from both sides. This will stiffen it up plenty and allow you to put more things in it without risk of it falling out the other side. Also it would give you a bit more width for storage than installing dowels on both sides. If you have a router, you could rout a shallow dado on one side and put in two squares of 1/4" plywood or just rip the all the boards, except for the front one a quarter inch narrower and to the same thing.
Modern wood glues are stronger than wood so yes you should be able to remove the castor, insert the glue and clamp it before trying again and this time predrilling the holes. I've done this many a time. The hard part is getting the wood glue into the narrow portions of the split. I use hide glue (which is thinner than say something like Tightbond) and work it down in the crack and clamp. Hide glue also has a much longer work life so you have plenty of time to do a thorough job and not worry about it starting to set up.