1 - It mixes functionality. CSS is for styling. HTML is for markup. Javascript is for functionality. This whole conversation regards separation of concerns. By inlining CSS or using React or whatever, you've now broken that way of thinking. Now, if theres a problem, theres not 1 files to look at for problems (the css file) but possibly many. It's a time suck.
2 - How do you deal with hover? or nth? or mobile / responsiveness? or a thousand other things CSS can do very well that'll take Javascript lots of code to accomplish? Sure - you can use JS to bind mouseover to a href and change it; but CSS does it so gracefully. These frameworks rely on JS so much to build the page; now it has to rely on JS to style the page also? Because CSS is old and inlining style is new and fancy? Inlining style has been around since the inception of html. This is not a new idea; it's an old idea that was thrown out for efficiency.
Inlining CSS is good for 2 possible scenarios IMO - Emails - which you really have no choice but to inline style - there are so many browsers / email clients / online email clients and so many of them don't support CSS, inlining the style is the only way and the preferred way to style an email. The 2nd scenario being an absolute quick hack. We've all done it - something is broken, doesn't look right, the boss is pissed and wants something changed; etc... do a quick inline style to fix it and deal with it later. But to build an entire website with inline style; I'd vomit in my mouth.
The article in question I feel is almost completely related to the fact that React inlines HTML / markup so the train of thought is - well, we're doing it to the html, why not do it to the style also? That's a very bad way of thinking about it IMO. Fix the reason why React needs inline html / style - don't through out the baby with the water.