Here is a little test I did (which is possible because I have two independent Signal accounts). I muted my extra account and then used it to send a message to my main account. From chat view, there it is right at the top with an unread count, even though it is clearly muted.
Either muting should prevent this, or an additional setting could perhaps prevent it. In the mean time, it was easier just to switch the family to #Threema.
And yes:
• Threema could have the same issue (they also have no 'full mute'). I am not claiming it is better.
• Yes, I am obviously aware how to setup multiple Signal accounts but it is non trivial and AFAIK not possible with iOS (my wife has an iPhone).