Sadly it does: in practice there are 5 big banks in an economy. Maybe less.
So let's say a bank has $10m deposits. You're right: they will lend out $10 million, most of that in mortgages, a bit in business loans, a tiny bit in personal loans. Mortgages come with the condition that you don't get the money, you get to write a checque (or do a bank transfer), verified by an official like a notary public. Now you might notice ... so wait ... they lend you money that you can only deposit into another bank account (and then very likely stays in that bank account) ?
So that means they can pull this: $10 million in deposits. They lend out $10 million, $9.5 ends up into accounts in that same bank and become deposits. So now they have $9.5 million in new deposits and $10 million in assets (money people owe them). Plus an additional $1 million in interests. Because there's only 5 banks where money can be kept, they're pretty much guaranteed, once above a certain size, to keep or exchange 1:1 deposits created from loans in their bank.
So now they lend out the new deposits ... and of course also the assets. So now they lend out a further $21.5 million. Next step is a further ~$40 million. In most of the world laws specify that they can only multiply ~50 times this way (Australia being a very notable exception).
So with $10 million in source deposits a bank would lend out a little less than half a billion $.
Obviously in practice it's 100 times more complex than described above. For example, banks will usually sell a good part of those assets, like loans, and then take that as a profit, and pay it out to shareholders. This changes their risk profile in positive ways. There's business loans which mostly (the larger amounts at least) work like mortgages but not entirely. There's international transactions, ...
The thing to remember is: large amounts of money simply don't "really exist". Starting at maybe $100 million it is simply not possible to "have" that amount of money. Something else is happening, some financial construction is involved.