No, this is untrue. You don't have to do all transactions in bitcoins - if it is accepted as world's monetary backing, you can still use paper money or electronic transfer inside one bank, the actual bitcoin transfers would be happening on an inter-bank trading. You won't be buying in-store using bitcoin directly - there will be an intermediate processor, just like Visa, which would aggregate thousands of micro-payments and send one BTC transfer.

If the transfer costs rise significantly, and blockchain grows, only big institutions with their own servers would continue digging blocks, and only large transfers would be financially viable.

However, that is perfectly fine, because no matter how many banks participate or how many different currencies are created, the backbone of it would still be valued in BTC.