What bothers me is this - if I were to withdraw my entire USD bank balance as cash and then hide it in a safe in my house, I would not feel secure about that at all; I would be constantly checking it, thinking of hiding places in my house to move the safe to, acting nervous whenever I left the house, etc.
You feel safe storing your USD balance in a bank because the banking industry is regulated, audited, and insured. If someone at the bank screws up and the cash at the bank is stolen, you can count on the government coming in and restoring your money to you. There is no similar concept in bitcoin. You are responsible for your own money.
Isn't a bitcoin paper wallet basically the equivalent of just storing cash in your house?
Somewhat, except that you can't create backup copies of your cash, and you can't encrypt your cash with a password. Of course, if an average criminal breaks into your home, and sees a pile of cash, it's pretty much guaranteed that they'll know exactly what they are looking at and be immediately aware of how much it is worth. If the average criminal breaks into your home, and sees a piece of paper with some letters and numbers written on it, they are far less likely to take it (unless they knew you owned bitcoin and specifically came looking for it).
So if someone were to break into your house and take the piece of paper, they would have access to all of your money.
Only if you didn't encrypt it with a decent passphrase first.
I also read that there is currently no way to assign a passphrase to a paper wallet.
Not true.
So if this is true
It's not.