I'm not sure I follow this. You believe the skimmer adversely effected calcium and alkalinity levels. And secondly, as a result you stopped skimming but advocate skimming as tried and true. I am sure I am missing something.
Regardless, in my experience, skimming does pull out a lot of gunk (some detritus but not really sure how much--I think socks do this pretty well though), DOC's ( I think 30% is the skimmers limit or so), and aerates the water pretty darn well.
But I guess the real question: is it necessary. I don't know. Thought sand was necessary a while ago. Is it possible to compensate for its absence, perhaps. I've run a skimmer and always viewed it necessary. But I've also thought water changes were necessary and there are those that don't.
So would the tank succeed with compensatory changes. Would there be an increase in the bacteria inhabiting the water column eating up all that extra gunk and DOCs or would the water just get increased level of nutrients and pollutants. Would oxygenation decrease and what would happen to pH levels. All of this is interesting but I don't think I would want to be the one to experiment. Keep us informed about what you do and how it works out