If I maintain a business bank account for my LLC in the US, and if I let the money I make from my business remain in the business account (without withdrawing to my personal bank account) and use that money only for business purposes, then do I still have to report that income and is the money left in the business account taxable?
An LLC is a "flow through" entity. Assuming you are the only member of your LLC, it's also a "disregarded entity".
What that basically means is that the IRS treats the LLC is if it doesn't exist: the assets of the LLC (including its bank account) are treated as owned by you personally, and the income of the LLC is treated as earned by you personally.
And everything else follows from that.