Sure, there is the possibility to perform counts of such metrics as visitors and visits still available, but overall a lot of insight has been lost. Despite the massive impact that the pollution of secure search has had, there is still some fish to catch, at least for now until the pollution gets worse.
SERP position analysis
The SERP position (i.e. search engine result page position) is still floating out there, and is actually possible to capture without using a SERP position checker tool.
Each position of the Google click through per visit can be reported upon and the results are interesting, for an average sized website the Google website position for every visit reveals an alarming detail that the Google click through rate by position it is evident that ~75% of the visits are related to the #1 SERP position (~25% for the rest of the Google referred visits).
Note that the share of total visits with the SERP position included in the example was at ~49% of all search engine related visits during the 8 week period of data, with increasing pollution this could drop meaning that yet another data source vanishes.
From a statistical viewpoint however that is an acceptable share to base conclusions on, especially if compared to the share of search term data from Google.
Replacing "not provided" with useful data
In order to better understand the distribution of clicks and move beyond simple search engine click summarization, using the page names (or better yet if available) page tag data can provide improved understanding of what was searched for. Page names typically contain information on the breadcrumb path and the page title, a page tag on the other hand typically contains words that identify what a page is about.
The data in page tags as well as search terms have previously been used to populate tag clouds, given that search terms are close to digital extinction in the Google sphere, page tags as such work out really well as a replacement for tracking down all of the "not provided" visits. As an alternative, page tags can be replaced by the measuring the Meta keywords of each unique page. This should provide an accurate data source on what users where searching for moving forward.