That is not quite correct, unfortunately. Since Matter over Thread communicates natively via IPv6, the Aqara M3 is perfectly sufficient as a Thread Border Router (TBR): it simply routes the IPv6 packets from the Thread devices transparently into your local home network. However, the statement becomes correct again if you replace “own TBR” with “own Matter Controller”. In your HA setup, this controller isn’t additional hardware, but rather the Matter Server Add-on. This runs in the background within a Docker container as a matter.js application (earlier versions used Python for this).
Since we were just talking about IPv6… you might want to check out my Matter & Thread Guide, specifically the section on Link-Local vs. GUA vs. ULA. That could actually be a reason why your devices are dropping their connection. In fact, if it happens again, it’s highly likely to be the issue. Feel free to send me a PM, and we can take a look at it together!