For the reference of anyone who comes across this to whom it is useful info, what happens for me seems to be as follows, on a Galaxy S21FE.
One note: The reason I can currently still access all of these steps is because my GSMT line went to Airtime Expired (due to their disabling autorenew) shortly after I last renewed my Nearly Useless Free Plan line (which was never able to auto-renew) in mid-January, I think it was, and by the time I remembered to come back and renew that line again to save it from closing in late March, the GSMT line had gone Airtime Expired for a whole month and therefore lapsed into Closed. That line offers me the renewal process and fails at the end telling me it can't be renewed because it's closed. To the best of my recollection, it all looks the same up to that point. If there are any differences that I'm not remembering, the following may be inaccurate.
I launch the app; it takes me to the last screen it was on previously and makes me watch an ad. If I'm on the screen for a line that needs renewal, there's a red RENEW button at the top. Tap that, I get another ad. After I close that ad, the bottom two thirds of the screen are a thing that tells me I qualify for the $10/mo plan and has the green button "GET STARTED" and the blue button "RENEW FREE PLAN."
Note: At least part of the time, it shows me this after the ad that plays when I open the app or switch the selector at the top of the screen to focus on a different line. However, if the line doesn't offer the RENEW button, the RENEW FREE PLAN button also won't be there.
The RENEW FREE PLAN launches a web view based on my default browser, as far as I can tell, and I believe the first time it did so I had to log in to that, separately from logging into the app. It isn't launching the browser itself, because I don't have any other tabs, but it tells me it's running in Brave. At this point I seem to be on a mobile version of their website, except it lets me manage my subscriptions, whereas the website no longer acknowledges I have any lines at all.
The fact that it pulls up a web view may mean that if either one has the permissions on their phone locked down particularly tightly, uses an unusual browser, or has a sufficiently old system webview (which might mean the app isn't fully compatible with older versions of Android), the click-through steps won't work correctly. If one is using something similar to Brave and has it configured to discard cookies after every browsing session, I would not be surprised if that also meant having to log in to the web view every time it opens.
Those are just some guesses, at this point. I could certainly understand if One is beyond the point of considering it worth any more attempts by now, though, so I don't know whether this will be of use to anyone.