I am also a frequent My Maps user.
On 16 September, I received an email I had been expecting for some time (based on previous announcements from Google about Maps) saying
Hi, We want to let you know that in the coming months, we're upgrading
the content you created in My Maps. All of the maps that you've
created will automatically move to the new Google My Maps (previously
called Google Maps Engine). Classic My Maps on maps.google.com will no
longer be available. You don't have to take any action to ensure all
your content will be upgraded.
The FAQ pointed to from this email says: https://support.google.com/maps/answer/6009908
It doesn't explicitly say that the Save to Map feature has been removed - but then it's really about new Maps, not My Maps Classic. Google's usual style would be to note announce feature removal like this, when the feature concerned is being replaced by another one.
Currently, if I go to maps.google.com, I am re-directed to the new maps.
However if I go to a website where one of my existing maps is embedded and click the "view in a larger map", I do go back to Classic My Maps, and am able to edit the existing map. However when I am doing so, and I search for a place, there is no longer a Save to Map option.
I am expecting that the ability to edit maps in My Maps classic will be removed sooner or later - and at that point I personally will need to upgrade one account to My Maps Pro to get one of the features I need.
Best Answer
Use Google Takeout to download the data. With KML format it can then be viewed in Maps by others.