Missing the deadline by a single day is a tough break, but it's the consequences that can occur with the system. It is very possible that some players will not be able to participate in Trials due to having active parasites, and even War. The simple solution is don't do anything that will result in getting them in the first place.
As for appealing to the Council for a temporary exemption from the bans, I don't see how this is a good idea. The entire purpose of the system is to create a culture in the PvP community where quitting Tournaments, Leagues, Events, Trials, War, etc. will not be easy and without consequence. Giving people a way to ignore the consequences doesn't really serve as a deterrant at all. Using your example for instance, what would make that player think twice about quitting something near War when he can just file an appeal and ignore the ban? Moreover, why in the world would War want players who can't keep their commitments? If you got banned for bad behavior, there is no way in the world you should get a free pass to play, and if you got banned for quitting then why should you get more chances when you have already demonstrated an inability to either manage your time, understand your limitations, or stick to your commitments even if you get bored? Everyone is told before signing up to make sure they understand the rules of the PvP function, and the PvP Parasite system. If someone feels they cannot live up to those expectations, then they should not participate. The rules are very clear and very simple and are there to ensure the best experience for ALL players, not to act as a paper tiger.
As far as appeals, you can always PM me or another PvPEM and request an appeal if you feel you were unjustly given a parasite. These are handed out by the organizers of each event, and they simply PM me or tell me in chat who is getting one and why. I trust their judgement so I accept their reasons without extensive research into each individual situation. That doesn't mean they are infallible so definitely contact me or another PvPEM and tell us why you feel you don't deserve a parasite. I can assure you we will look into the matter and if we agree with you we'll reverse it.