I believe you can set your voiceover rotor to move forward from text to text. Since almost everything in the Apple News Plus app is a link, moving forward by text field or whatever might just move you forward to non-heading headings much faster. Try experimenting.
Next, if you open an article and go to the bottom, I believe you will find the options we used to be able to get by flicking up or down on an article link. Share is near the top of each article, though.
You can also simply remember on what page your desired section of the news app can be found. Mine, politics, is found in the vicinity of page 14, so I flick up with 3 fingers until I get to page 14, and then I start hunting with flick left and right.
Finally, don't forget the item chooser. You might find it can speed up your hunting quite a bit.
I agree Apple's changes to the news app are certainly annoying. I've reported it to Apple accessibility. You might want to do the same in an email. Squeaky wheel and all that.
Someday, Apple will wake up to the wonderful idea of changing their operating system to allow users to define a heck of a lot more, like where things are, how they are displayed (e.g., text versus heading), etc. I would love it if they did that to the menu bar, too, and to the Apple menu where you find about, system preferences, app store, etc.
Hope this helps.