The crux of the problem was that the JRE didn’t have the Java Cryptography Extension (JCE) Unlimited Strength Jurisdiction Policy files installed. The solution to fix JCE-related issues (as evidenced in Burpsuite’s Alerts tab) was to search for the term “Java Cryptography Extension (JCE) Unlimited Strength Jurisdiction Policy Files 8 Download” on which led to the download page for JRE 8’s JCE Unlimited Strength Jurisdiction Policy. If you’ve managed to eliminate the JCE error message in Alerts but you still see an SSL handshake error, try resetting Burpsuite’s SSL options in User Options > SSL > Java SSL Options.