A password manager because if anything goes wrong, you’ll be completely screwed.
What you SHOULD absolutely self host though is a password manager, so you can be in control of your most sensitive data.
Regarding email, I think everyone should absolutely self host it, but it’s less and less viable in this google/Microsoft duopoly world. But ideally everyone would self host it. The reason why people advise against it really comes down to lack of real competition, and the two tech giants dictating how we violate every RFC possible.
Basically yes, you can install the app. Where is the data for that app stored? If on the phone, then your data is lost when your phone dies (maybe even when you switch phones). If in the cloud, that’s great, but then someone else has your data. Selfhosting is to get the convenience of “the cloud” (multi-device/user sync and sharing) without actually relying on the cloud.
Take Jellyfin/Plex/Emby for example. Yes, you can just connect a laptop with and HDMI cable and play from VLC. Or you can just use an USB stick in the TV and play from the file system. But you have 2 TVs, a tablet, and a phone that all want to watch something from your movie collection. Of course you can just plug this stick into other devices, and use it that way. Or subscribe to Netflix that may or may not have the movie you want to watch. But what if you had your own Netflix that can be used by multiple phones, tvs and tablets, even at the same time? What if it even synced the progress of a series so your TV no.2 knows where you left off on phone no. 4? This is what self hosting is made for. You HOST “Netflix” yourSELF.