What business is it of the government what parents choose to screen for? Even more pointedly, how can one support elective abortion and not support genetic screening for whatever the parents want to screen for?
Note: The following is for a world where genetic screen is cheap and can screen for basically anything that is influenced by genetics. That's way more than we can do now, so if your question is about whether it should be the government's business now then never mind. :-)
It's not clear that it makes sense to compare abortion policy and genetic screening policy. People consider abortion to control the number of children they produce or the timing of their production of children, whereas they would use genetic screening to control the types of children they produce.
As for why the government might have an interest in what people can screen for, consider what happens if people start screening for aesthetic reasons. That could be strongly influence by fashion.
You could get a generation where most children were selected for some specific body properties because that's what some big star had. Often bodily features favored by fashion aren't actually good for the person who has them.
For example suppose there we have a few years where extremely thin people became the fashion rage, and people who want children during that time start screening for babies that will be prone to anorexia or other eating disorders.
I can see the government having a legitimate interest in shutting that down.
> As for why the government might have an interest in what people can screen for, consider what happens if people start screening for aesthetic reasons. That could be strongly influence by fashion.
I don't think the government should have much say in people's fashion or aesthetic choices either. I'm sure that eventually designer babies will be a thing and when it happens parents will follow fads, but fads will come and go. A generation or two of more skinny people isn't going to be a problem as long as those people can continue to reproduce and odds are good that we're going to get even better at helping people have children when it's hard for them too.
The vast majority of parents will always favor having healthy and beautiful children no matter how fashionable sickly looking people becomes.
The government's interest wouldn't be in the fashion choice. It would be in how the fashion choices are implemented.
A generation or two of skinny people who are skinny because their parents used screening to only have children who have naturally thin body types should be fine from the government's point of view.
A generation or two of skinny people who have naturally average or heavy body types but are skinny because they have eating disorders which result in them eating much less than they need to remain healthy should be of concern to the government. It could cause serious problems for the health care system.