The reason that John is this way is rather easy to figure out. When we see him react to gentle kidding with an act of malice and violence, we immediately see him for what he is: a vain simpleton who can't tolerate being laughed at because he's the only person who really exists. Since other people aren't fully real to our boy, he can't really let them be happy lest their happiness be at his expense. Also, he likes his unquestioned and unearned authority so having happy, smiling children means that he ain't got them under control.
Elly is also rather simple to figure out; not only is her natural state one of pointless gloom and unreasoning negativity, she's insanely jealous of the younger brother who had life so easy and was so happy to, as she saw it, rub her nose in it. When you factor in the fact that Elly will not ever get over all the stupid crap that happened growing up stupid under the Red Ensign, she doesn't see an innocent child who isn't trying to ruin her day when Mike, Liz or April listens to the hated voice who tells her that this life is worth living. She sees the Phil who she saw as telling her that she can never win at life.
This need to make sure that Mike, Lizzie and April realize that life is Serious Business and that fun is for suckers is why they never laugh in front of the children if they can avoid it. To them, it would be an act of cruelty to let the children know that they have senses of humor.
The end result of all of this is a Michael who stands there panicking because his children want to play with him and a Liz who is about to freak out because the Weird Frenchy Girl engages in harmless wordplay. Since this allows Elly to crow about her own misery and to make them hopelessly dependent on her for advice, this is a great thing.