Since Joomla! 3.5.0, released two months ago, utf8mb4 is a core Joomla! feature. If your server supports it, updating Joomla! will switch your database to utf8mb4. You should not use the Admin Tools feature on Joomla! 3.5 and later (including 3.6).
Please keep in mind that for utf8mb4 to work you need to have a new enough version of MySQL, PHP, the MySQL connector libraries and the PHP-MySQL connecting code too. If one of your sites doesn't convert to utf8mb4 automatically it's just because the server doesn't support it. Nothing us or Joomla! can do about it (trust me, there is an entire 6-month-log saga in the Joomla! issue tracker about all the work half a dozen people did to make utf8mb4 work as smoothly as possible).
Nicholas K. Dionysopoulos
Lead Developer and Director
🇬🇷Greek: native 🇬🇧English: excellent 🇫🇷French: basic • 🕐 My time zone is Europe / Athens
Please keep in mind my timezone and cultural differences when reading my replies. Thank you!