What does it tell you to do in your Ops Manual or equivalent?
What does your Chief Pilot / Flight Technical Manager say on the subject?
In my experience, for boxed quick-don masks, pre-flight test is to hit the test button and ensure the flow indicator shows oxygen flow.
A general visual inspection of the removed mask and inflatable harness, operation of the quick don harness, operation of mask mic and cleaning of the mask and stowage is carried out on scheduled maintenance at a period specified in the Aircraft Maintenance Program. The AMP is derived from the Maintenance Planning Document published by the aircraft manufacturer and approved by the NAA in the country of manufacture. Many "experts" are involved in writing the MPD looking at failure modes, whether the failure would be evident, impact of the failure, etc., then decide on what inspections / tests should be carried out and how often to ensure continued serviceability of the system.
How to carry out the tests and inspections is detailed in the Aircraft Maintenance Manual.
Could a damaged mask or harness be deferred - I doubt it as it is safety equipment but if you have read the MEL, what does it say?
Is the quick donning function important - yes, otherwise why would we have it. Does it need testing every day, in my opinion, no, what could go wrong with it on a day to day basis, packed away in its stowage? The AMP will call for the required inspections and tests at the appropriate time to maintain serviceability. Pulling the thing out of its stowage every day and having to get an engineer to repack it the correct way will probably cause more damage than good and a waste of time and money.
I would suggest you speak to somebody in your maintenance planning team to take you through the AMP and AMM to put your concerns to rest.