"Plus the restriction of its application to elite competition undermines the whole policy."
I agree, but the IOC can only put in place rules for the Olympic games, they don't get to tell other sports federations what they can and cannot do.
Having said that, the World Athletics rules (from last year) are basically the same - although there is no reference to CAIS.
World Aquatics (swimming & diving) and the International Weightlifting Federation both have a rule about not having gone through male puberty.
However, anyone competing at that level who wishes to aim for the Olympics will, of necessity, have to do the one-time cheek swab or saliva test.
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But, as you say, it is the lower level competitions where things may be different. But it may not be all bad.
Here in the UK, British Weightlifting changed its rules in July of last year, just after FWS, to limit participation in the female category solely to biological women (they had previously had a testosterone limit rule):
"To comply with the Equality Act 2010, BWL will only offer a female category which only includes women and girls whose biological sex is female. BWL will not permit the participation of males who identify as female.
"Furthermore, no male has the right in law to self-identify into a female changing room, toilet, or other single-sex space in accordance with the Equality Act 2010."
England Athletics also has a similar rule and organisations like the National Athletics League also follow this position.
But this doesn't apply to eg schools events (unless at a national level) or individual club events.
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In the USA, President Trump has had a big effect. About a year ago he made an Executive Order that said all sports bodies that receive any federal money should have a female category based solely on biological sex.
From there, the US Olympic Committee told the different US sporting federations that they also had to comply and these federations are moving at different speeds towards this. So, some still have outdated rules but they are all moving in the same direction.
But again, very small or schools events may not be covered by this.