I do the same (over in comp sci.) - usually the journals will let you put pre-print articles up on arxiv, researchgate, and web pages, and then the final article - sometimes with added spelling mistakes - many of the journals have off shored the editorial process - is on their site.

I'm curious what would happen if a new paper tried to reference previous papers published in ArXiv instead of the traditional publishers. From a strictly academic perspective, I can't think of any reason why this would be a problem, apart from the existential problem it causes for traditional publishers.

Citing papers which aren't yet published by their arxiv numbers is routine. (And, before 1991, people did the same using the author's institute's preprint numbers.) If they have appeared in a journal by the time yours is accepted by a journal, then you should add the journal info too. (Although many journals now allow you to leave the arxiv numbers as an additional part of each reference.)