The setting is that of a world with a technological level around the Interwar Period and a societal level around that of the Victorian Era.
Though answers to this question are encouraged to go back to the Renaissance
All good and well, this question is about the decision of a region/country-spanning clerical organization to accept the burden of providing and maintaining medical- and welfare facilities & services.
They are one of the oldest institutions in the region/country and thus have in their possession a great many estates and lands which provide them with ample money.
They are exempted from tax-paying duty to the governments they exist under and their institutions are neutral-ground where theirs is the only jurisdiction.
They do not receive any monetary compensation or other benefits from the government(s).
They maintain their own universities where doctors and other medical staff are taught and trained.1
They have to provide medical services for military casualties in form of providing room and services in places with military academies, as well as maintaining sickbays in the field and at barracks.
Q: What reasons are there for such an organization to take over these burdens/tasks?
The mentioned burdens/tasks have likely not been taken over all at once but have probably grown and cumulated over time.
The question is looking for answers that provide valid reasons/explanations on how this has started as well as why it has continued
1These often overlap with clerical schools, but that does not really have any influence on this point.
Addendum: The church itself is a rather huge polytheistic confluence of older simpler polytheistic and monotheistic religions of different areas and tribes.
They started off as the spiritual part of herdsmen and nomads of the Green Wides. Due to the nature of the people of these areas, the original churchmen started creating fix settlements where the wandering groups would find shelter, could exchange wares and bury their dead without fearing bandit raids or similar.
Over time the wandering groups would strife further north, west, south & east and thus get in touch with other groups and religions, which they understood to integrate into their own pantheon in order to mingle with these others.2
2Others that soon became part of their growing culture & society.
This section has been copied from the predecessor/inverted question found here