To go with examples not child related.
People who have survived a life-threatening illness or event often say they appreciate life so much more afterwards.
They are not saying their life is more important that someone else's life, or even that they appreciate their life more than people who haven't been through something similar, they are saying they personally appreciate life more than they used to because of the experience they went through.
They are saying that they realise how close they came to losing their life and the experience has changed them, they are different in many ways and one of those ways is an increased appreciation of their own life.
They may also be saying that the event was so big that they view things now as before the event and afterwards, and can see how markedly they have been changed by it.
Maybe the woman in the OP's post would have been a much less anxious parent if she hadn't waited so long for her child. Maybe she would have felt exactly the same as she does now.
I suspect I might have been less anxious if we hadn't been through loss. Yes we knew we could conceive. But staying pregnant was difficult, required operations and interventions and we still lost two babies, I nearly died, and we nearly lost DS at the 11th hour of my pregnancy with him. We didn't know if we would ever manage a full term pregnancy and even if we did we were not guaranteed a child who survived it.
That has an effect on people, it changes you in ways both good and bad. I would have been a different parent if we hadn't had those experiences. Maybe not a better or worse one, but a different one. Perhaps I would have the same concerns but not to the same extreme as now.
Things like this, loss and infertility, can hold a magnifying glass up to your existing fears and insecurities and make them grow, but talking about that doesn't mean you are saying other parents don't share the same worries, or have less to lose.
Basically it's just saying "I wouldn't be this way, except that this happened to us, and now everything seems that little bit more frightening to me that it did before and it's changed me."
That is all the woman in the OP's post actually said. She admits she is an overly paranoid parent, the reason being because after eleven years of trying to conceive, which included two rounds of IVF, taking risks with her child that some other parents judge to be acceptable or which they don't consider as risks at all, is just too frightening for her now because she does have that experience behind her.
I can't imagine that she wants to feel that way, or enjoys it, but it's what her experience has brought her to, and that's all she really said. She didn't say her child was more precious or special, she acknowledged her own fears and explained why she has them.
The rest of the OP's post was made up in the OP's own head.