What is the point of publishing? I think the answer most would give to this question would be 'because I have to.' You have to publish because a CV or resume full of publications allows to keep your job, apply for a new job, become a professor, and get tenure. I feel this myself and this need has become conflated with the desire to publish my research because I want to share it with the world. Pragmatically though I must regularly publish if one day I hope to see myself as a tenured professor.
Another answer a more career-secure researcher would give for publishing is 'to share research among scientists.' This is the reason journals were created so that large paper books could be distributed to universities. I am old enough to remember going to the library to physically photocopy the journal articles I wanted. A more specific question is then, given we have the Internet and many journals have become online only, 'What is the point of publishing now?'
If a group of scientists were to come together to create a solution for sharing research would the current situation be close? That we spend weeks formatting research into Microsoft Word, or LaTeX if you're lucky, for third-party publishers that will, after months or even years, eventually decide to disseminate to everyone. If our research is declined we then have to repeat this process for a different third-party publisher. Once this research is finally accepted many publishers will then charge scientists to finally get access to it.
This pay-for-papers situation means that many poorer institutions cannot access current research because they cannot afford the monthly subscription for the articles. You might think that perhaps these poorer institutions are mostly in the third-world. However if you have ever had to ask a colleague for a PDF then you are working at one of these poorer institutions. I think the list of non-poor institutions that have access to every journal subscription a researcher could need is very small or perhaps even 0.
You could argue then that open access publishers are fixing this situation. Open access journals charge researchers for publishing an article but then make the article free to access for everyone. There are still, however, limitations to open-access publication. For instance can you perform large scale analysis on the huge amounts of scientific data these open-access journals contain? Can you access structured citation data for these open-access journals?
A good question might be "who needs access to this large quantity of data?" Which scientist needs the citation information for all published articles? The answer is almost no one. The reason is the same for why most people don't need to download the entire web to their computer. You only need to download the web pages you're interested in. One group that does however need to download the entire web is Google. Their algorithm searches every web page to see which pages are linking to each other. This algorithm then identifies the top search results because they have many incoming links from other pages. Unfortunately there is no equivalent of Google for scientific publications. So while I can search the web an optimised search algorithm I am stuck in pre-Google days for searching the world's research. All because the research is siloed across many individual journals.
A solution to the slow publication time of journals is to use pre-print servers. This means uploading your research to a different third-party that will make your research available to the community until it is more formally published in a journal. This situation means we have acknowledged that the journals so slow at publishing that we must an additional third-party to take up the slack. If we are becoming more happy using pre-print servers to share research doesn't this mean that publishers are a becoming a vestige from a pre-Internet period?
You could argue that publishers provide an important a role in screening research through peer-review and journal exclusivity. However both of these functions are performed by academics rather than journals. So if academics are performing the necessary peer-review why do still need the journals? Is the lack of will to organise ourselves a good enough reason to put up with all the inefficiency, effort and costs that publishers bring?
Do we expect that the current status quo for sharing scientific research will still be the same 20, 50 or 100 years from now? If the answer to this is no, then why do we settle for it now?