Accounting books is what makes that impossible. Any financial institution must keep accounting books rigorously to GAAP standards.
If they accept your money into their account, then they must make a bookkeeping entry for it. It's a double-entry bookkeeping system, so they must make two entries: where the money came from, and where the money is going to. By "where" I mean which internal account in the accounting sense. There is a ledger of internal accounts, everything from "janitorial service" to "interest on Loan 12344678". As a bank, each customer will have an account as well.
They can't make a lopsided accounting allocation or their accounting system won't balance internally. They can't make no entry or their external bank accounts will not balance with their accounting system.
Any company of any size has their accounting books audited by an independent third party. They'd have to do a pretty good job hiding your money with smoke and mirrors, and that sort of thing will be turned up in a careful audit. Doing that thing is how you get to do "the perp walk" on national TV while an army of SEC agents haul Bankers Boxes out of your building.
If an agent told you to transfer money into their personal account, they'd go to jail, simple as that. It would be a very easy crime to prosecute.
Anyway, a competent discount brokerage sets up accounts like a bank, anyway. Your account will have an account number that is all yours (just like your bank) and a 9 digit routing number. If you have any doubt, ask your banker to bird-dog the transfer: they will look up the routing number and confirm it is really that broker, and may be able to confirm with the broker that the account is yours.
Regardless, half your trouble is that you're trying to do this 'the new fangled way'. I just don't have problems like this: I write a paper check to myself, and endorse the paper check as follows:
FOR DEPOSIT ONLY
Account of Harper
At Fidelity Bank
Fidelity account # 1234-5678
Routing number 123456789
use those exact numbers ;) J/K
If someone handling that paper check did something else with it, that's just plain old-fashioned check fraud. Nobody is going to say "oh, that's an accident or a technical glitch". Paper works. Paper holds up in court.
Even if you sent that to the broker's house, the broker couldn't do anything except deposit it in Fidelity as instructed.
Of course under Check21 they will scan it into an image, but still, everyone in the chain of custody will take one look at the handwritten scan and say "yeah, there's no mistaking the writer's intent with this one". That will make fraud an open-and-shut case.