Application removes itself from "login items"
Yes, you can change it by buying the Pro version of Matthew Palmer's Vanilla which will allow you to permanently add the app to the Login Items list.
If you really do not want to give 4.99$ to a guy giving out free software that he spends his own time making, you could startup the app using launchd or similar that doesn't use Login Items. Or you use a similar method to re-add it to Login Items when it is removed.