I don't think any of the recommendations here are even close to what OP is asking for... QUIK from my understanding is just a replacement SMS app, it does not "sync messages to other devices" or allow you to send SMS messages via your phone from other devices, nor does it have a desktop/web version, all of which is what Message+ does. Pretty sure this requires a self-hosted server to do (or a third-party proprietary service like MightyText).

SmsMatrix, KDEConnect/GSConnect, Nextcloud Talk are some examples that will do this.