I refinanced my house and tried to have the cash-out direct deposited to my savings account at Barclays. It was denied because hidden on a back page of their website it says that Barclays will not accept any wire transfers of any amount. No reason is given.

From the bank's general FAQs:
Our routing/ABA number is 031101321.
Please note: Barclays does not accept wire transfers at this time.
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From the bank's general FAQs:
"Your external account is a bank account you have with another U.S. financial institution. It’s linked to your Barclays account so that you can transfer money between the two. Any external account that is linked to your Barclays account must be in an account holder’s name."
2. Choice wrote: "ACH to a checking acct? Bet they would do that"
I'll take Choice's bet (and I might lose); for the amount that the original poster wanted to wire.
Barclays is an online bank that, as far as I know, does not offer checking accounts. So, I don't think Barclay's would allow the original poster to ACH the money to a checking account.
But maybe I'm wrong. Or maybe it could be ACH'd to a Barclays checking account that was opened back in the twentieth century when Barclays had branches in the U.S. that offered checking accounts. I don't know if any of those accounts still exist.



3. What is an "external account"?
Your external account is a bank account you have with another U.S. financial institution. It’s linked to your Barclays account so that you can transfer money between the two. Any external account that is linked to your Barclays account must be in an account holder’s name. This includes payment processing companies such as: PayPal, Dwolla, Coinbase, Venmo, Square, and Payday loan deposits.
4. Tell me about "test deposits".
For security reasons, we use test deposits as part of the external account verification process. 2 to 5 business days after you add a link to your Barclays account, we’ll make two test deposits of less than $1.00 into your external account. We do not use your money to make these test deposits.
We’ll then ask you to log in to your Barclays account and report the amounts of those test deposits. After you have confirmed the test deposit amounts, you’ll be able to begin transferring money between your Barclays account and your external account. We use this method to verify each new external account. If you do not verify your external account within 99 days from when you opened your account, transfers will be restricted until you can verify your external bank account.
https://www.banking.barclaysus.com/faq.html
I have no specific knowledge, but I'm skeptical about your assumption. I would not draw any conclusions based on having received tax refunds or stimulus payments.
P.S. and yes, carly, it's possible that you have conflated various methods of transferring funds. Wire transfers are one such method, but not the only method.