I have recently switched plugins for website contact forms to Ninja Forms. The main functions are free and they also backup emails in a database plus lots of extra paid-functionality if needed.
This also meant I had to find a new way to integrate Google Analytics to continue tracking where leads come from.
Tracking hits from buttons is called “event tracking”. Setting this up has always been niggly and frustrating in my experience but this solution is very elegant. It’s not important to know how it works, just that this method makes it quicker and easier implement.
In addition to general website traffic data, tracking button hits from contact / newsletter / registration forms will give you a deeper understanding of how your website is performing.