Update: solved.
I tested with a clean local install of WordPress 4.4.1 and TEC Pro 4.4.3. This combination caused the identical problem as we are currently having on the production site.
After updating this clean local install of WordPress from 4.4.1 to 4.7.3 TEC automatically started working for recurring events.
We have yet to perform this update on our production site but every indication is that a WP update will fix it; it does not appear to be a PHP version issue. If we discover otherwise we will let you know.
Many thanks for your help.