It has been done. For example, a case I know is $prestigious_journal = Lingua, $prestigious_journal1 = Glossa. http://languagelog.ldc.upenn.edu/nll/?p=22162

And there have been some more cases, e.g. in mathematics (a field where single authorship is common, see below why this can be relevant).

But why is this the exception and not the rule, at least for now? I can venture several contributing factors (probably not the whole picture, as it's a quite complex issue, but I think these are at least among the most influential factors):

1. As an academic, I can tell you that a universal characteristic of academics (and even more so of those relevant enough to be in prestigious journal editorial boards) is that we are starved of time. Most of us are already quite overworked and there is no way in hell we are going to find time to inform ourselves about web journal publishing systems and set one up.

2. To officially be a "prestigious" journal in some countries, including mine, you have to be in the ISI JCR impact factor list. And that list is for profit, among other requirements journals have to pay to be there. So there is the additional hurdle of figuring out how to get the money.

3. Even if you can solve points 1 and 2 (which by the way could definitely be done if universities or even states got involved, providing support for open access journals), there is still the problem that establishing "prestige" (impact factor) takes time (a journal takes several years to be indexed). During this time, your community will lose a journal to publish in and get points in your CV. For a senior academic this might not be a big deal, but the problem is that senior academics typically work with PhD students, postdocs, young professors looking for tenure... and for these, the major journal of their specialization suddenly imploding and having to wait 4-5 years for the new one to get an impact factor can be a huge deal for their careers.

Believe me, the overwhelming majority of academics aren't happy about the status quo. After all, we are working for free for these journals, spending hours and hours of our time writing papers, polishing them and especially reviewing, only to see how they reap the profits and even try to charge us to read our colleagues' papers. I don't know any academic who likes that. But in the hyper-competitive arena of academia, where not publishing enough papers can pretty much mean the end of your career (and if you have reached stability, you almost always have other people that depend on you whose career is at stake) they pretty much have us by the guts.

In my view the solution has to be top-down. It has to come from university boards and governments, who are the ones controlling the incentives that make publishing in profiteering journals pretty much mandatory. Set up public open-access journals maintained by universities or states; change the requirements for hiring, tenure, etc. so impact factors issued by private companies are not taken into account and to incentivize publishing in said public journals, and the problem would pretty much be solved. But there doesn't seem to be the political will to do it (although at least the EU and some universities, as seen here, are pushing a bit towards it, which is good news).