I've brined chicken for the first time today, and I included garlic and ginger in the brine. My question is, could I take the garlic and ginger and add it to my cooking later after draining, or should I throw it out?
I'm concerned if they become flavourless after brining or oversalted.
You should not. Brine at the concentrations for brining meat does not kill all bacteria, and you had that ginger and garlic swimming together with raw meat.
The main issue here is that you asked about removing items from the brine to use for another time. This is a problem, as they have been cross contaminated, and it would give the bacteria a chance to multiply.
Even if you were to kill those bacteria, you don’t know what byproducts they created, so you don’t know what’s required to denature them.
If you really wanted to try to make use of them a second time, I would recommend immediately boiling them to make a stock, and then straining and refrigerating the stock (discarding the solid matter, which should have given up its flavor).
If you cook it immediately, e.g., as part of the dish it would be fine. If you want to save it for later use, as already answered, it would likely be bad due to cross contamination. In theory, you could flash freeze it, but that is hardly worthwhile.
