This may sound harsh but it's meant to get through that your behavior isn't ok either in Riot's eyes or the communities since Tribunal would've punished you in a heartbeat ahd this case come before it.
The fact you queue up knowing you've got family interferences that will cause you to miss time means the issue isn't with Leaverbuster but you. You choose to play knowing you're about to screw over up to 9 other peoples game because you can't plan around your time constraints. The system is fine and you didn't just suddenly get punished out of nowhere but got punished because you not only used up all the built in leniency the system has but continued to leave/afke even after you got punished.
You don't get a get out of jail free card just because you behaved for a few games either, you've got to earn that right back by behaving for quite a bit more games played without leaving/afking/dcing etc. I've had my power go out, my internet die, modem die, firewall crash bug etc as have many others yet in over 3 years or so have never even gotten so much as a warning let alone a ban. I've got 2 sons who're teenagers now and I've had to plan around them growing up when I wanted to game.
Many people have Wifi that goes out, real life emergencies that popup etc yet they never get banned either because they don't use the built in leniency up because it's not a common occurrence. If you can't dedicate the time to play a game then don't queue up unless you're willing to suffer the time penalties up to a actual timeban.
The reason the penalties are much harsher is because Riot wants to get the message across much sooner that you have to stop leaving/afking/dcing so much and either dedicate the time or don't play until you do. You can play custom bot games, co-op vs ai just fine as long as you're doing it via the invite button and not queuing up via the in game queue system.
If you think 20 mins is long just think of how many games you've made your team lose and how much time those people wasted because you chose to queue up knowing you were going to leave the game. No one likes to 4v5 and the other team is usually not happy winning a game because you left. If you want to be able to play then arrange with your parents times that you can play uninterrupted/do your chores/homework etc ahead of time.
When you can manage to do that then queue up and play but until then go the custom route so you don't ruin everyone else play experience and you don't keep getting penalized.
