Either I need the app to forward any received text message to another phone number. This should be reliable, i.e. not stop when the phone is idle/sleeping.
Or, I need to sync all received message to a cloud, Nextcloud, Google Account, proprietary; so that I can read message from another device. This needs to be reasonable secure, i.e. either e2e encrypted or stored to trusted service. I’m willing to trust Nextcloud or google for the purpose. I’m also willing to trust a third-party service when run professionally, but I’m sceptical.
I like the SMS gate app from f-droid suggested.
A different approach to read SMS from other devices, for example your PC, would be using kdeconnect. It’s supposed to allow you to read current messages, receive messages, and reply to messages, even send messages I believe. So perhaps an option, though not a syncing tool to keep SMS messages somewhere else, and I can’t tell about the encryption involved…