You're talking about the population dropping by 4-5 billion, which implies that the death rate should double. There's not much evidence to think this will happen.
I depends on what your definition of evidence is. Yes, the world population is still growing, by about 75 million a year. Right now there are no global pandemics brewing, and starvation is still confined to the usual areas. However, if you step back and take a look at trends -- in energy supplies, food supplies, rainfall pattern shifts, glacial melting, the implications of financial collapse for foreign aid, social disruptions etc. -- the picture isn't nearly so rosy. I understand that trend projections are probably not the "evidence" you're looking for, but for those interested in understanding the range of possibilities we face, the trends are assuming more and more relevance.
Resource depletion and environmental degradation are enormous problems, but this isn't the first time we've faced them. there have been resource depletion crises in the past, long before the advent of oil or industrial scale coal mining.
In past cases of resource shortage, there were still other places on the planet we could either move to or get resources from. This is no longer the case. Mankind is now fully globalized, and virtually all accessible resource bases are being exploited. There's nowhere else left to go. Even the search for new oil supplies has shifted to tar sands, deep water and the forlorn hope of the Arctic basin, all of which are high cost in economic, environmental and net energy terms. The same thing applies to resources as diverse as copper and potash.
Environmental degradation has reached the point where CO2 is melting virtually all the world's glaciers, 90% of the big fish in the oceans are gone, and much of the surface of the Pacific is covered by man-made plastic detritus. We're at the absolute limits of what the planet can provide and absorb. In the past when this happened we usually moved on. Now there's nowhere left to go.
Your argument assumes that things such as wind or tidal power etc. etc. are doomed to failure.
I don't think they are necessarily doomed -- we're going to be deploying them like mad over the next couple of decades. However, their success in keeping the good times rolling is by no means assured. These technologies face significant problems in terms of capital requirements and the fact that they require significant fossil fuel inputs in the construction phase. Capital is draining out of the world economy at a frightening pace, and there is enormous competition for oil supplies, just as the production rate may have plateaued.
This doesn't mean the technologies are doomed, but if they fail to materialize as planned, the consequences will be frankly enormous. My assessment is that the probability that they will not materialize as planned is a lot higher than their proponents are willing to admit, and the consequences of that failure would be so severe that we should at least be considering the possibility.
I can't take anarcho-primitivism very seriously
The problem with any label is that as soon as you use one people assume they know your position. I use A-P as a descriptive philosophy, not as a prescriptive one. I think people like John Zerzan, Derrick Jensen, Daniel Quinn and Charles Eisenstein (though the latter doesn't call himself A-P) present a very coherent analysis of how and why humanity got into this predicament. Their observation that the human presence on the planet was sustainable up until a few thousand years ago is persuasive to me, as is their critique of how things have gone wrong since then. That obviously implies that I think things have "gone wrong" and indeed I do.
I part company with them when they get prescriptive, assuming that because we were sustainable during the Paleolithic that getting us back to that state would solve our problems. The arrow of time is unidirectional, and we will go on from here -- you can't go backwards. There are lessons from the past that we can take forward with us, though. Those lessons include the idea that humanity's relationship to nature is one of partnership rather than ownership, that egalitarian societies tend to be better than hierarchic ones, and that we should be very cautious about anything that encourages either a separation of Man from Nature or the elaboration of our hierarchies.
As far as using the Internet goes, I have no problem with that -- it's part of my ecological niche, along with automotive transportation and government bureaucracy. As I said, I embrace A-P as a critique of civilization, but I reject its prescriptive aspects that might imply I shouldn't be a fully participating member of my society. It's a secular philosophy, after all, not a religion. I try not to let myself be defined by the symptoms of civilization, and accept that they are as impermanent as everything else in the human experience.