Hi,
Many people figure it's best to keep on top of day-to-day hypertension by taking all their blood pressure medications in the morning.
Since hypertension alreadys mean that the blood pressure is high enough, why do they think it is best to keep it on top?
I have underlined the relevant parts of your post. The sentence uses the phrase "to keep on top of something" but you have queried "to keep something on top". They are completely different.
You need to keep on top of your homework assignments = you need to keep up, you need to make sure that you deal with them in a timely fashion, you need to make sure that you don't fall behind with them.