Considering how ubiquitous plastics are, they look rather harmless to me.
We are wasting hundreds of millions of tons per year, we eat it, drink it, we have been doing that for decades and we are still discussing statistical signifiance.
Sure, not completely harmless, but few things are. And considering the tremendous value plastics have brought to society, I'd say the downsides are comparatively small. What worries me more than human poisoning the article is about is the effect of microplastics on marine life, as it is less studied than people and there is some evidence that the physical (more than chemical) properties of plastic particle might be damaging.
It doesn't mean there isn't something to be done. Proper waste management is the obvious one, and I don't mean anything fancy, I mean "do not empty you trash in nature" thing that is unfortunately much too common, especially in developing countries. Properly managed landfills would probably solve 90% of the plastics problems. Reducing disposable plastic is, I think, generally a good thing, but not before making sure the alternative is not worse, and of course making sure that "disposable plastic", is not "throw it in nature plastic". We should also research better plastics: make sure the durable ones are actually durable, and that the disposable ones degrade properly, also consider the petrochemical industry that backs it up and maybe switch to more sustainable sources.