Drain cleaners work from long exposure on a blockage. You wouldn't want to handle a chemical that could dissolve all that hair and bio buildup by pouring it through a slow drain because if it splashed on your skin it'd burn through your skin in the same time.
Manual removal is always best. If you want to be proactive, get a long test-tube brush and put it down the drain periodically so you don't have to disassemble the drain.
It's disgusting, but it's satisfying to take it apart and give it the old toothbrush like you suggested from time to time.