Impossible to synchronize the SOLR index from the admin

All you several options but not the one that would be by far the most useful which is to call the index job which is used as startup.

Also the option would deserve more documentation since it's either false ("Add missing" index whatever you give it to index and never test if it already exist) or not very clear what it does ("reindex" would deserve a big warning indicating that you are going to delete your entire index and recreate it from scratch).