That's a matter of preference, what you're already used to. But the way that I personally do things when it comes to the design part is first do the "desktop version", then the "mobile version", i.e. I don't do those things simultaneously. If you have a dual screen setup, it sounds useful to have the DevTools on one screen and the browser on another, is all I can tell.
Here's the reasoning behind "first desktop, then mobile": first, I start with the problem, and it takes a lot of time to get things straight; then, I try to build a mockup of how the product (website) should look like considering all of its features (elements); then, I write the HTML/CSS. Here, I am only focusing on the desktop, because my focus is turning the mockup into a reality, into the finished product. And everything else that is related to "mobile" or "tablet" or whatever smaller-screen-view comes with the use of CSS media queries. So that's not the main problem, should not be the main concern.
Once I am done with the "desktop mode", my goal becomes making that to be responsive and adaptive. Responsive, in this case, means some elements are stretchable/can shrink upon resizing. Adaptive means that layout changes depending on the size of the screen. And I am absolutely using the mobile mode from DevTools, it is not a "jank", it is actually very useful, because I can choose from the dropdown what screen I want to emulate, including toggling the touch mode, or adding some throttling (also an underrated feature, because most people don't really care about performance), etc.
Hope this answers your question!