"After each intermediate upgrade step, the Moodle service failed to start"
...I'm unsure what you mean. Surely, if you'd upgraded from 3.8 to 4.0 and it failed you would stop there. You wouldn't take additional steps? I'm not sure why you're heading for version 4.4, as this is (effectively) unsupported.
As you have to do this in more than one step, let's look at the jump from 3.8 to 4.0. Please confirm that you read the release notes (https://moodledev.io/general/releases/4.0) and that your environment meats all the requirements?
If so, what exactly were the errors? Especially the PHP errors.
Do you have a test environment to try this out on? If your system is critical, then you should...
(See also, http://www.syndrega.ch/blog/#php-and-dbms-compatibility-of-major-moodle-releases)