[0] https://www.bitsaboutmoney.com/archive/two-americas-one-bank...
Banks try to avoid holding excessive amounts of cash in suburban branches because it makes them attractive armed robbery targets. If you have a need for a large amount of cash, you can let them know (my bank says 24-48 hours in advance), go to multiple branches, or to a large city branch. Inconvenient, perhaps, though temporary. If it's for a business or a recurring need your bank is generally happy to make that part of their regular logistics, "Monday's cash delivery needs X because asimovDev regularly wants $50,000 in cash every Wednesday".
> it has to be approved
I'm not sure what you mean by this, but I don't think it's "approved" in the sense of the bank deciding if you're allowed to do so. But hmm: multiple ID checks, or a specific seniority of bank employee doing the check. Or if you decide you want to take out $250,000 and you have to do it in cash, they might a) want a day or two to physically acquire the cash, and/or b) have additional security due to their insurance policy. They certainly might try to suggest you look at a cashier's check or wire transfer or other instrument to do so. There's going to be a CAR for any transaction involving more than $10,000 in cash.
But I'd challenge the assertion that the bank tells you you're "not allowed" to withdraw "large amounts of cash" from an otherwise unencumbered bank account.
People I know who have $250k in cash or cash like assets generally keep it in a mix of dollars, euros (one paranoid guy I know has a bunch of Canadian dollars - no idea why), junk silver, gold / silver bars (for the former, you're talking about $150k just for a 1kg bar), a collection of watches that appreciate in value, and so on. To quote one guy, "I want to make sure we can wear my family's wealth", which they indeed did (for no good reason, they're just from a cultural background that leads to them being paranoid about banks) when they moved from Scandinavia to Australia - watches, gold bars, etc. on them and their children's persons, valued easily north of $300k in current money.