"Peak Oil" is a theoretical model for analyzing oil supply and production cost. It was first conceptualized in the 1950ies, in a time where hardly anyone was claiming that we're almost out of oil. Indeed "Peak Oil" does not claim that the absolute oil reserves are nearly empty but that the reserves that can be exploited cheaply are at their maximum production capacity. What this model does not take into account are price fluctuations, based on a decrease in growth and economic activity (what we have seen in the past 10 years) and new technologies (Fracking, LNG). So this model is imprecise and incomplete, but useful when analyzing trends over a very long period of time.

You claim, that the "Automation crisis" isn't a real problem either, because apparently employment is almost back to pre-2007 levels. This may well be true, but a significant factor that has to be analyzed is: what jobs are being taken. I have no numbers only anecdotes [from UK/Europe, not US], and there is a pattern where low paying service jobs are replacing jobs in the manufacturing industry. This is a direct consequence of automation in the manufacturing industries, and these jobs are not very secure / automation-safe either. Also, this problem is way older than you or me, the first Luddites were smashing machines in the 1810s. The history of the 1800s show that automation (with lots of other factos in conjunction) can lead to societal upheaval and severe economic problems.

As for productivity growth, I concur that by the numbers in your first link show a stagnation in the past decade. That does not mean that this trend will continue in the future. When AI/ML Technologies become mainstream (this is already the case) and receive widespread adoption in several industries (this is not the case by now) productivity growth will increase once again.