Kaspersky suggestions re changing settings didn't work for me, as I already had those settings.
After various setting changes, I did manage to fix the problem by doing this:
1 Open the desktop Kaspersky then open settings on lower LHS; 2 Find Web Anti Virus section and scroll down and open Advanced Settings, then scroll down to URL advisor; 3 move down and place a tick by "Check URLS"; 4 move down and click on "all websites except those specified"; move down to "Checked URLS" and click on All URLS.
This restored the green or grey icons to search result lists, as I wanted, and extended them to some websites I use which is ok with me. The green icons are smaller than the lost search list ones. Having safe searches back is a big relief 🙂
I'm surprised that a computer simpleton like me found a solution that worked after Kaspersky's suggestions didn't.