Yes, at the point where you try to withdraw actual cash the bank will have to use any cash it has as its assets, but nowadays the majority of transactions are digital and all that needs to happen is to adjust the balances of the payee and payer accordingly (potentially this can involve the interbank settlement network but the end result is the same).
Once you stop thinking of money in terms of tokens and bank deposits as those tokens getting stored in individual buckets - or there being any buckets at all - you can think of it in terms of assets and liabilities it makes sense. Also that commercial bank money (deposits) and central bank money (cash) are different because they are liabilities to different entities - commercial banks and the FED respectively.
Or more fundamentally, modern money is just I-O-Us being moved around.
https://cepr.org/voxeu/columns/banks-do-not-create-money-out...
https://www.federalreserve.gov/econres/feds/money-reserves-a...
