I would mail the 2022 and 2023 now, then wait a couple weeks before mailing the 2021.  Otherwise the 2021 refund might be issued (although I think IRS freezes a lot of refunds for past-due years, when later returns have not been filed).   It's true that you don't know a 2021 refund will be issued, or that tax is not owed for earlier years.  But it's also true that the 2021 refund might be frozen for many months, while 2022 and 2023 notices demand immediate payment.  Bottom line is that your client is violating the law, and it's not your job to tell him how long he can ignore it. 

Of course, we're all assuming that the government won't close down Saturday.  If that happens, I would still want my returns at the front of the line when IRS reopens.