The first common feature comes to us from the notes; that's because Lynn said that she had never actually had to send a child to hospital with an illness. Since she isn't really all that good at imagining what people do when she never did it, we end up with parents who sit home like uncaring idiots while their sick, frightened child wonders why she's been abandoned. This is like how April thought she was going to end up locked in a dungeon because Lynn couldn't imagine John and Elly actually talking to her. In both cases, Lynn never actually bothered sitting down and wondering what people would do and decided to have them do what she would. Since she doesn't seem to understand the social norms that apply, the result is rather off-putting.
The second commonality proceeds logically from the first; this is because we have Lynn not really understanding what it looks like when Elly acts as if doing the bare minimum to qualify as a decent human being is some sort of heroic sacrifice. It's as annoying to think that Lynn wants us to praise Elly for being a hero for sitting in the waiting room while Lizzie still cries for Mommy as it will be when she declares that Elly "understands' April when she tells her that she's not upset about what she's upset about.
Finally, we have to contend with the fact that Lynn has a strong aversion to having to care about people in genuine need. Whenever the conversation veers towards the subject of sick relatives, Lynn tends to lavish praise on people who don't ask her to feel bad for them. There are two reasons for this appalling tendency of hers. The first is that her own fear or being weak and dependent tends to make her think that people who are weak have somehow committed a crime and thus aren't worthy of her attention. The second is that said awful people are trying to impose on her and make her weaker than they are when they have no right to do so. This means that Lynn has criminalized compassion and empathy because she wants all the sympathy for herself.
The end result of all of this is that she never reused this sort of storyline because she did not get the results she both expected and wanted. Instead of getting letters lavishing sympathy on John and Elly for having to take care of some stupid kid who went and got herself sick, she ended up with a lot of letters that angered and confused her about how sad they were for Lizzie and how angry they were that John and Elly weren't doing enough.