On the grand scale, over time carrying capacities will generally rise as the crops and animals used are improved, people will get better at living in cities, merchant and priestly casts will keep getting smarter, farmers and herders will keep getting more resistant to infectious diseases and better adapted to their diets. And this should accelerate as long as population keeps getting larger.

Carrying capacities may increase, but so what? You don't see a whole lot of innovation out of Africa. Per capita is what matters, and per capita there is no long-term upwards trend. To quote Clark's Farewell to Alms:

The wage quotes from 1780–1800 do seem to confirm that technological sophistication is not the determinant of wages. English wages, for example, are above average in the table, but not any higher than for such technological backwaters of 1800 as Istanbul, Cairo, and Warsaw.7 English wages in 1800 on average were about the same as those for ancient Babylon and Assyria, despite the great technological gains of the intervening thousands of years.

On a time-scale of what, a century? In the absence of oil and coal I can't imagine say 18th century European civilization stagnating on a time scale of millennia. These are cosmic eye blinks. Canals, wind-power and watermills had already basically created a mini industrial revolution in some places. I also can't imagine say Aztec civilization stagnating in the absence of easily obtainable gold or copper.

I disagree. 18th century Europe (by which I mean, of course, England, as the starter of the Industrial Revolution) was already dependent on non-renewable energy, and had been since roughly the late 1650s. Look at the charts in http://www.voxeu.org/index.php?q=node/6781 and notice where 'coal' exceeds firewood+water+other-renewables. (Firewood began falling in absolute terms; Easter Island comes to mind.)