Why in the world should a single branch be expected to hold that kind of cash for withdrawn without notice? There is a* huge* difference between "having the money" and "having the money in cash form on location".
They're not autonomous though - individual branches don't hold the value of all their customers' accounts as cash. You're subject to a daily limit (though you can take more if you give them notice).
Now we're just being pedantic. My point was that each branch should have enough cash to act as a "proper bank", whatever amount of money you'd reasonably expect one to have in its vault.
If they barely had 10k in their vault, then it logically follows that they almost never need that much (otherwise they would obviously make it a point to keep more). If they rarely need that much, then it logically follows that somewhere at or less than 10k is a reasonable amount to have. There is no sense in taking the increased liability of having more if they don't need to.
Furthermore, acting as a "proper bank" means they do a lot more than just acting as your personal piggy-bank/mattress. If they have all their money tied up in cash, it means they can't actually be preforming real bank activities (investment).