I find it very annoying that every time I am saving my webmap I have to go back to the settings and configure my parcel layer as search layer. Even if I didn't change anything. This is a registered bug since 2015(!) , please do something about it, Esri!
versus
Strange, I haven't noticed this behavior with the web maps I've done this on...does it only happen for certain web maps for you or all of them?
I don't recall this happening before. I just tried the same with a different subscription and the settings were not removed. Very strange. I will keep testing...