I was going to suggest the same thing. If you build out the device yourself, the privacy is as close to perfect as you can get. Take it up a step by having a dedicated hidden and non-internet connected WiFi network for your doorbell/security systems. A few disks connected to the Pi and one of their cameras should be sufficient. I'm sure there's pre-built cases and button enclosures you could pick up.

Bonus: You can build the device to your local privacy regulation. Where I am, doorbells in shared living spaces can't have audio recording, but good luck finding a decent doorbell that does not also record audio.