Overly complicated and inconvenient. The same thing can be achieved with a smartphone and an Internet connection. I don't know who in their right mind would run to the bank to deposit cheques and receive digital tokens in return. It's usually the other way around.
I am talking about making offline transactions possible (I see that you didn't learn to read)
In fact, I didn't think it all up. What I described in the OP was a real process how gold certificates came about. People deposited gold (now it would be Bitcoin) at an interest in a bank, and they got certificates confirming that they actually deposited so much gold there. Since these papers were payable to the bearer (i.e. they didn't have the depositor's name in them), they were used as a payment of debts between people and later got used as paper money for the exchange of goods
I get it, but first, the chance for banks to adopt bitcoin is too small. They have different interest to do/manage (fiat money).
If in the future, banks decide to adopt bitcoin and manage it as they did on gold, print paper bitcoin and hold the private key.
It sounds nearly impossible and too complicated, like u said, they could manipulate the number of bitcoin that they have.
Well, I don't like the idea of paper bitcoin. Physical bitcoin such as casascius, denarium or titanbtc seems more interesting for me.
Read here