For me, the problem was that I had optimized my search index by removing everything in the Included Locations from my Indexing Options except Start Menu. I did this because I use a different desktop search which I find far superior to WDS ('Everything' from http://www.voidtools.com/). So I wanted to avoid the redundant indexing and keep Start search limited to just Start related things.
This gave me the exact same symptoms as in the poster's question - installed apps show up fine with a Start text search, but all the control panel and system settings applets are missing, plus the constant "We're getting search ready".
I actually had this problem in Windows 8 and 8.1 and hadn't looked into it because I was using Start8 and it only manifested for me using the search charm, which I rarely used. Plus I figured Windows 10 would fix the issue. Obviously it didn't. Upgrading to 10 preserves the WDS configuration. Being a pre-10 issue also rules out Cortana being involved in this particular problem - a ton of the "fixes" I've seen online involve doing various things with Cortana.
I resolved the issue by adding my user folder to the Included Locations set and waiting for indexing to complete.
If I rebuilt the index with and without my user folder included, I could see the bad Start search behavior go away and then come back. Verified 100% that this was the cause for me.
If anyone from Microsoft is listening: it would be nice to have the special "Start Menu" entry also include whatever magic bits Control Panel etc requires that including my user root picks up.
Also a quick note: if you want to completely reset your WDS folder configuration, you can go to HKLM\SOFTWARE\Microsoft\Windows Saerch, set SetupCompletedSuccessfully to 0, and restart the Windows Search service. Checking and unchecking boxes can end up with annoying overrides of inclusion-inheritance that are apparently impossible to remove otherwise.
I spent some time trying to isolate which folder it really wants, so that I can exclude my user folder, but then include just what it needs. (I don't really like including my user folder then excluding all the folders in it, because then I'll have to go exclude new folders as they're added to my user root for whatever reason.) But I wasn't able to figure it out.
It's either a hidden folder or possibly one of the junction points in the user root. The default setup excludes AppData, and all the junction points in my user root point into AppData, so it would seem that none of those would be visible to WDS, yet something is being indexed.. And the UI doesn't permit selecting any of that stuff in or out so it's hard to experiment with.
So I gave up. I now have my user folder included and then unchecked every folder in it. It's not "pure" but it is a ~2600 item index, and the problem as reported by @mejobloggs is completely fixed for me. Good enough for government work.