Interesting that you have 2000+ apps. I thought I was the only one! 🙂 I wonder if that has anything to do with the issue.
I have the same problem on iPad and iPhone, including having erased my phone again and restoring. I had the same experience on the betas on my iPad and reported it as a bug. So I thought it was the expected behavior when I saw it on my devices in the released product. But in posting my question elsewhere in the Community, a person helping me shared a screen capture showing his Search working correctly. I don’t know how many apps he has, but you raised a good point.
I’ve been using the App Store option at the end of the Search list, and when it takes me to the app that I needed, it has the little download cloud next to it, even though the app is already on my phone from before, and I’ve already used it since upgrading. Or I will swipe through the “dots” to get to my last desktop more quickly for only 1 more swipe to get to the App Library Search.
I reported this issue in the feedback app. I hope enough others will report, for Apple to investigate a fix.