Because yes, it does sound pretty ridiculous that an LLC needs to maintain a $0 balance or else it will penalize the owners of said LLC. That doesn't sound right.
It's definitely not correct at all. Neither is issuing phantom invoices to yourself to put money into the account.
A bank account is just a vehicle to store business cash, it really has nothing directly to do with the way you should be maintaining your LLC's books. You should be recording things like that as "capital contributions" which increase your "capital account."
Similarly, profits and losses should be booked to your capital account. I would strongly recommend doing some reading on capital account bookkeeping.