I recently did exactly this. I'm heading to DefCon next week and the Classic is a nice hands-off high security device I can use for just about my email and little else. Did the same last year with an older 'Berry as well and it helped me to enjoy the conference rather than my phone.

Email still works without carrier/BB support for push email?

Yes. I've been using a Blackberry Classic for over a year now ("upgraded" from an old Nokia Symbian phone). Does plain IMAP (TLS 1.2), with IMAP IDLE support just fine. Only annoying "feature" of the email client is that it insists on sending HTML mail by default, to send text/plain you have to set it per message/thread.

To add, the old reliance on this was because older Blackberries (<= OS 7) needed to be assigned to a BIS server at the carrier. With OS 10, they work differently. There are core BB 10 servers that are set to be shut down Dec 31, 2019 though, and it's not really certain what the devices will be like after that point.

I bought one on ebay for 40 bucks. Beautiful phone. Super solid build, and design - definitely, turns more heads then your everyday iPhone X. If it wasn't so laggy (didn't take a millennium to boot-up), I would use this phone always. Also, it side-loads android, so I got Snapchat, Spotify. Slow, but they are there. The blocker on this phone is the primary interaction method (the keyboard), and that's why I found myself away from it. Just turn off the notifs. I guess I didn't need a "dumbphone", I do need one where it's not convenient enough for me to just swipe up and read on for hours.

I had to carry one for work purposes when these were popular. I found that the 15-minute startup time is pretty much negated by the three-day battery life, and the phone was faster than my android at actually running apps.