I am running 1.2.43 on Linux Mint Mate 19.3. I have designated an address to open when I start Brave. Sometimes it does and sometimes it does not. If I close and restart Brave it may open on the correct page but often not. If I poke around in the settings edit (but do not change) the address it might start correctly. Currently it refuses to start on the desired page no matter what I have tried.
Another anomaly I see… An icon relating to the last site I accessed is shown on the “home” page. This is a concern as it implies that the information was stored somewhere and not cleared when Brave was shut down.