I abandoned the Links Page plugin for good. I replaced it with the Web Links plugin. You can see it in action on this site.
Web Links plugin was originally an UnNuke plugin (forked from PhpNuke). Because I use e107 as the backend for my still active UnNuke sites, I decided to create e107 version. If it detects the Links Page plugin is in use, it copies its data to new needed tables.
They removed the Links Page plugin from the official repository too. I removed it from my GitHub account, and you can find it now only in the repository with old plugins.