I'm using Safari on a new Mac mini. I've updated zotero, installed new connector. I do not see the Zotero icon to the left of the search window, nor anyplace else on the page.
The toolbar button doesn't appear and disappear. It's always present when the Zotero Connector is installed, and the icon just changes depending on what you're viewing. (You may be thinking of the behavior long ago from the original Zotero extension for Firefox, where an icon actually appeared and disappeared in the address bar.)
If you don't see the toolbar button at all, the Safari extension likely just stopped showing up due to a Safari bug:
It was not showing up in Safari settings. I deleted, reinstalled, and it seems to be working fine again. I switched to a Mac thinking it would simplfy my life. Hmmm.
Upgrade Storage
If you don't see the toolbar button at all, the Safari extension likely just stopped showing up due to a Safari bug:
https://www.zotero.org/support/kb/safari_compatibility#can_t_see_the_save_button_save_button_flickering
Many thanks for your help on this.