Can you save an advanced activity search?
The key functionality to do this was added in CiviCRM 4.7.25, with CRM-20630.

My understanding is that it's almost but not quite what you need.  This allows you to pass variables through in the Activity Search URL, but it won't generate the URL for you - you'll have to figure the URL out for your search.  A extension to auto-generate the URL seems like a worthy project though.

The feature doesn't seem to be documented yet, but you can see examples on Github.