The recent email about the Adobe hack was well-intentioned, but it unnecessarily worried a lot of my clients. None of them have Adobe accounts, but they got the email because my email address is also listed on their account (and it was part of the leak). For some odd reason, I didn't actually get the email (at my affected address), I only heard about it when clients started forwarding the email to me.
Another instance was during the brute-force attacks on WordPress sites – my clients were unaffected, but the email they received sounded pretty alarmist while still being too technically complex for them to really understand.
I don't want to resell hosting services; I prefer that my clients have their own DreamHost account so they can contact support, pay their invoices etc. However, when it comes to technical, scary-sounding messages, it would be great if I could specify an address those should go to, so they're not sent to my client's email.