Will I get charged for an iOS app that a family member already paid for, if I download normally?
Short answer is, it depends on whether the app in question has Family Sharing enabled for it by the publisher of the app.
If the price shows up, even though a family member has already paid for the app, does that mean I'll actually get charged that price again?
You won't get charged for the app in question if and only if the following conditions are met:
You are a part of a Family under Family Sharing.
A family member has already bought the app in question.
The app in question has Family Sharing enabled for it by the publisher.
Or is this simply a (giant!) bug in Apple's App Store app?
This is not a bug from Apple, it's by design. Not every app on the App Store automatically becomes available for free to the rest of the family if at least one of the family member has bought it. It's up to the publisher of the app whether to enable Family Sharing for their app or not.
So it would appear logical and consistent that you as a user would see the app normally along with a price tag on the purchase button (which is an expected behavior). When you attempt to buy the app by tapping on the price button and it meets the eligibility criterion for Family Sharing, you are prompted that a family member has already bought the app, and, because it has Family Sharing enabled by the publisher, you won't be charged for it.
(P.S.: While it's for fact that this behavior is by design, the reasoning behind logical and consistent appearance is based on my personal interpretation.)