That’s exactly what’s happening. The States that ended the extend benefits have lower unemployment rates. Ex: https://apnews.com/article/business-health-alabama-coronavir...
Your “example” is of a state that had a lower unemployment than the national average before ending extended unemployment benefits seeing not significantly different than the national rate of decline in unemployment in the brief period since cutting extended benefits, not an example of more rapid drop after cutting benefits.
You'd be better looking at rates of employment.