Geologist here! The report in question [1] and supplement [2] seem pretty think-tank-y, but I would say they pass the sniff test to first order (if we have a biologist or ecologist around, they could probably say better). The focus on animals makes it relatively plausible -- these are charismatic and easy to observe, but a lot of the ecological heavy lifting / biogeochemical nutrient cycling is really carried out by microbes, which are are probably doing better. Microbes also represent most of Earth's genetic diversity, and will pretty certainly survive any mass extinction we throw at them.
In any case, the Guardian article is correct in claiming that
>Many scientists believe the world has begun a sixth mass extinction
This is pretty well accepted among geologists and paleontologists, as far as I've seen. Here's one relatively recent perspective on the issue, which covers some of the back-and-forth: http://science.sciencemag.org/content/360/6393/1080.2
[1] https://s3.amazonaws.com/wwfassets/downloads/lpr2018_full_re...
[2] http://awsassets.panda.org/downloads/lpr2018_technical_suppl...