For me the answer was "those pesky custom domains!". To expand on it - we have a custom domain assigned to the application. And to renew the SSL binding I had to:
1) go to "Custom domains" section, delete the existing custom domain assigned to the app;
2) add new custom domain with the same details as previous one;
3) configure it and add ssl binding.
A bit annoying that you cannot do it from the SSL settings section, that there are no guidelines on why the button is disabled, that you have to delete something and potentially cause downtime.
Whelp... Not going to say that my solution was the correct one or the best one.