I don't generally have a problem with this. I use JSX at work too. My take is it's just another templating system, I don't think it's special. I don't have anything against templating. To a certain extent, my point is that template systems are good -- put your two-way data bindings and special components in them.
When I write JSX though, I make sure it renders out as semantic HTML. If your approach to JSX is, "I'm going to write my render code in JS and output HTML", I don't have a problem with you, you're doing great. If your approach to JSX is, "my interface is just Javascript, so it's fine to spit out poorly organized divs that are absolute-positioned everywhere", then I think you've missed the point of HTML.
React isn't an antipattern. But React doesn't remove your need to think about the final HTML that your app is going to spit out. Apps like Youtube and Twitter are particularly bad at this -- Youtube's DOM is a horrifying mess, it's completely unreadable and way too difficult to work with or style. It's not because they're using a component system to build it, it's because they're fine treating the HTML as a non-readable render target. They don't care about HTML as a user interface.
The broad mistake people make on the web is to look at HTML and say, "because I can't build a scalable app on JQuery and hand-coded HTML, therefore HTML is bad." Google's hot-take with HTML custom components is not that you could put your template logic inside your other component logic -- it's that the HTML layer shouldn't matter at all and you should stop thinking about it.
Separation of concerns is for users, not for you. It's not about where you organize your logic, or even if you mix up your languages into the same file. It's about what gets spit out onto the final page.
> We’re killing CSS classes and building styled-components.
I would caution a bit against doing styles in JS -- not because adding a template layer or JS layer that generates CSS is bad, but because many codebases that I've worked with use CSS-in-JS as an excuse to rewrite basic CSS controls like hover effects in Javascript and inline the remaining styles. That doesn't mean you shouldn't do CSS in JS, it just means you should take the time to use a library that spits out actual CSS and that allows you to take advantage of native selectors and pseudo-elements. React does itself a disservice by using tutorials that teach people to apply styles directly to elements instead of pointing people towards any of the more decent companion libraries that will let you do styling correctly.
Unless your engineering team is the size of Facebook's and you can't communicate about your components at all, you probably should still be using classes. Of course that doesn't mean you can't use JS to restrict class scope or template that CSS. I personally tend to be more of a proponent of BEM than I am of JS styles, but I'm not going to fight someone over that.