Hawking is expert in a few areas. He's not an expert on population growth, ecosystems etc.
19th century biologists weren't experts on physics but they still showed Kelvin up. Pasteur wasn't an expert on medicine but he revolutionized the field. Pauling started out studying quantum mechanics but his contributions went way beyond that. Alvarez wasn't an expert on paleontology but he still dragged the paleontologists kicking and screaming toward the current consensus.
We don't have exponential growth any longer . . .
Strictly speaking, if you draw your graph with small enough intervals on the time axis, AT THE MOMENT, you are technically correct, in that the global human population growth RATE has been declining recently. But if you use larger intervals on the time axis, this disappears and you are back to exponential growth, and from a more long run perspective, I very much doubt that the momentary trend will stabilize and become the long term trend. All of human history, which is marginally relevant, and all of the theory of natural selection, which is totally relevant, is against it.
It is to be expected that if net reproduction rate is not rigidly constant (and absent some artificial constraint, there is no reason for it to be, and historically it hasn't been) that the curve of population against time, given a short enough interval between data points, will go through periods when the growth rate is less than, and periods when it is more than, what a constant exponential growth model would predict. But ANY long term net reproduction rate > 1, even 1.0000000000001, leads in the long run to a catastrophic increase in population that will force the net reproduction rate down through poverty, famine, warfare, disease or some equally unpleasant mechanism. I'd much rather we stop short of that so we can all afford the luxuries of biodiversity and a pleasant standard of living. If the rate fluctuates, as it naturally does, there will be intervals when the rate is less than what a constant exponential model would predict and periods when it will be more. The greater the fluctuation, the longer interval you have to look at to see the long term trend.
It is fashionable among most of the left, the religious right, and my own libertarian fellow travelers, to dismiss Malthus, but their arguments usually strike me as like the chap who has fallen out of an airplane and reasons that he has already fallen a long way and nothing bad has happened yet.
Larry Niven and Steven Barnes articulate the point very well in their novel "Saturn's Race", which I strongly recommend. I believe Eric Drexler makes a similar point in the seminal book on nanotechnology, "Engines of Creation", which I regard as one of the most important books of the past century. If net reproduction rate is lowered by arguments appealing to altruism, by rising living standards, by increased life span, by economic security, by lower childhood mortality, or any of the other influences usually touted, what traits are being selected for? Resistance to those very influences of course. Unless we very actively do something to prevent it, the future belongs to the descendants of the fecund, and it is a pretty grim future.
The Good Doctor wrote most eloquently on the subject. I regard his essays published monthly in F & SF as his finest work. He wrote that they were the thing he had the most pleasure in writing. Indeed, if you'll accept, for the moment, a broader definition of "SF" to include this sort of non-fiction, which I think of as the "penumbral literature of SF", and which I also associate also with Analog and with James Baen, I think they are among the finest things in the whole field. His essays in that series on human population growth are extremely worth reading. The intellectual quality is much higher than in the dumbed down essays he wrote for many, more pop, markets. Asimov adapted his writing to different markets more facilely than any writer I know of, and if you haven't read some of that specific essay series, I'd argue that you've never the real Asimov.
I fear we won't address this problem effectively until we are forced to, until biodiversity is reduced to people and people's crop of food yeast, possibly even until we've mined the Oort and turned all the available phosphorus into living tissue. We won't get there in giant leaps but by tiny increments, none of which will seem so obviously bad at the time, but which cumulatively lead to a dystopia fully as bad as any Huxley imagined.
exponential growth . . . anyway stops for before system failure in most domains.
That would depend on your definition of "system failure". For any definition likely to be applicable to the present issue and likely to make that a true sentence for the same issue, there are many ways the system could stabilize that any sane person would regard as far from optimal, without constituting "system failure" from some narrow technical perspective. There are all sorts of likely dystopias reachable through a Malthusian path that are short of extinction.