NOTE: My notes are for the United States, though there will be many similarities with other industrialized countries. And non-industrialized countries.
This is not unique.
Capturing and persecuting disabled children (in other words, putting them in institutions) was a very common response to "difference" in most of the 20th century and also some times before.
I know people who were institutionalized (with the approval of their families) for having Down Syndrome or juvenile arthritis (which doesn't affect intellect, just walking and using your arms). Even children with something as simple as club foot ended up in institutions.
Kids who needed physical adaptations to their environment (places to roll their wheelchair...or the wheelchair in the first place) were institutionalized when they were too big to carry around. Changing the location of the kid was easier than changing the infrastructure.
Mental health issues also were something that frequently pegged people as "unfit" for society. This included women who spoke out against sexism, people of color who were too "uppity," people suffering from trauma (especially things one didn't talk about, like being raped), and people with actual mental illness.
Historically, there were not a lot of people with autism in institutions. Simply because autism wasn't very common. Now it's extremely common (plenty of studies show that only a part of it is due to changes in diagnostic criteria and the increase in diagnosis overall). But if we were still a society with large amounts of institutionalization directly due to disability, the institutions would be filled with autistic children and adults too.
So if your fictitious society targets autistic children, it may simply be because autism is perhaps the most common condition in children with disabilities that aren't overly physical (blindness, losing a limb, etc).
We still have group homes and the like but they are not institutions and they are not for small children. Those kids who can't be with their families mostly go to foster care. But, guess what? We still have capture and persecution of people with disabilities in the US. It's called prison. Huge percentages of modern day prisoners are people with mental illness (often masked by self-medication aka drug abuse) or people who don't "fit in" in some manner or people with other kinds of disability.
With American-style prisons for profit, there's an incentive to increase the prison population. So, if you're part of a marginalized community, it doesn't take much to get sentenced. Then you're put to work. You get paid almost nothing while the prisons take payment from the companies who benefit from the cheap labor.
I'm talking about real life here. I'm not making this stuff up.
So if your world has a lot of people with autism and a lot of oppression of people who are different, then people with autism will be targeted. The only thing you have to do differently from real life is the fact that autism was not common in the 70's and 80's and is not common in 3rd world countries.
Adding to my answer as at least one person finds insufficient evidence of persecution:
What I describe above was not about lack of resources but was a systematic attack on anyone who was disabled. All of this was part of the eugenics movement (which did not originate in Germany in the 1930's, though they took it to new levels). In Nazi Germany, people with disabilities were rounded up and murdered. These murders largely took place before the concentration camps were fully set up. Institutions were emptied as people were massacred. People with mental illness and visible disabilities were taken from their homes to be killed.
In the US and many other countries, women with disabilities (especially anyone with an intellectual impairment or mental illness) were sterilized without their consent. In many cases, without their knowledge; they sometimes found out after years of unsuccessfully trying to have children with their husbands. Women of color with disabilities were especially targeted.
This was straight up eugenics. Lots and lots of documentation about this. The leaders of this movement were not secretive about what they were doing or why.