The Present Simple vs The Present Continuous

My father takes some medicine so he feels quite drowsy.

This sentence sounds weird to me, and rightly so.

Using the present simple tense with a verb often means something that's always or generally true. "I play the piano" means something like "I know how to play the piano" or "I enjoy playing the piano" which are actions/states that generally won't change.

My father is taking some medicine so he's feeling quite drowsy.

This should be correct. This means "My father is now taking some medicine.", where "now" can be any stretch of time including the present. The father could have just started taking the medicine yesterday or today, or will stop taking it tomorrow. It's currently true, but the action won't last forever.