Matrix is not proprietary. The protocol is FOSS, Synapse server is FOSS, Dendrite server is FOSS, there are FOSS clients, Element is FOSS too afaik.

Thanks, uninstalled

Are you certain that Element does not run in the background? It always does for me, both on my Samsung with OneUI and on another Samsung with LineageOS. Perhaps not really helpful, but my observation is that Element’s background “listening for notifications” is quite reliable. Might this be due to some settings in your OS?