Since your role is 'squeaky wheel', suggest to the server techs the use of 'git' for maintaining moodle core code.
Both contributors are correct in what they are advising about PHP versions/etc and hops.
Didn't really mention how. Personal opinion ... the 'standard/documented' way *not* using git is more prone to human error ... especially when one isn't that familiar with updating/upgrading Moodle ... which is evident in that your site is currently 3.0. If your techs use git ... much easier to update/upgrade core code - thus avoiding the issue of waiting too long!
'SoS', Ken