How can I fix composite deck railing that wasn't installed with brackets?
Without seeing a picture it is hard to say.

My experience with wooden furniture is that when you put it outdoors bad things happen, especially if screws and nails are used to hold it together. It is pretty rare to see a deck older than 10 years that does not look like a potential termite habitat.

In general, putting corner braces under the railing would certainly provide a lot more support than a screw through the top. Unless you bolt the corner braces you will still have the screw deterioration problem. This can be mitigated by using brass or stainless steel screws. Bolts, of course, tend to have that Frankenstein look, but at least they hold the wood heap together.

My guess is that one to two years from now we will start getting questions about decking deterioration from you.