Such a society could probably make new computers, but the cost of making new, much worse specced computers, may not be cost effective for a long time as long as old computers can be found and repaired.
Edit: such a movie or novel would make for great Fallout-ish world-building. Imagine for instance a large hotel in a big city running the booking system on an old laptop in a central secure location, and new 8-bit computers used as terminals at the front desk. Public access 8-bit computer terminals at public libraries connected to copies of WikiPedia running on some old gaming rig and such things. :-)
Post global nuclear war reconstruction would really benefit from having even a simple spreadsheet program to aid with resource allocation, wouldn’t it?
During a nuclear war reconstruction scenario, computing wouldn't be seen as ancient or unchanging. We'd be in a transitory period where we try and recreate computing infrastructure with what we have and the power sources we have access to. But we wouldn't develop an ancient tradition or anything.
A. For obvious reasons, we continuously improve tooling and processes. This creates circular dependencies that make our civilization non-restartable.
As a toy example, imagine we made tool A, which let us create tool B, which let us, among other things, upgrade A to A'. A' can still produce B, but can also produce upgraded B'. B' obviously lets us make A'', but also C', and A'' + C' let us make B''. Rinse, repeat for a hundred years. The result is that the whole industry relies on tools A'''''''''' and B''''''''''. The hobbyists still play with A and B, maybe even A' and B', but there's nothing between A' and A'''''''''' in actual use or production, nor anyone who remembers their design, because they've all been made obsolete long ago. If the calamity hits, and we can no longer make new A'''''''''' and B'''''''''', we don't get to downgrade a step or two - we go back all the way to A' and B'. And all the other tools and technologies that also relied on our advanced capacity - they all stop working and go "back to beginning" in lock step.
B. Making and operating tools involves ever increasing energy use.
C. We've used up all easily-available high-density energy sources long ago. Both our renewable tech and our fossil fuel mining and combustion processes are A''''''''''-level tech.
With that in mind, it's not that big of a stretch to write a plausible backstory for a culture as described in the article.
And if things are so low tech, the long-distance communication would be far more impressive than the story suggests!
It seems almost certain that climate change is going to severely disrupt several of these cycles in decade or two, and a slightly different failure mode of what you allude to in C. is that our tech and processes become capped before we adequately scale up renewables, and overshoot kicks in and it's actually a negative feedback loop that certainly doesn't just stop with computers.
Sometimes someone manages to spin up new manufacturing of retro components or things to replace the retro components (there are a few pcbs running around that can replace floppy drives that conform to certain interfaces with usb sticks or sd cards or even a network): Vacuum tubes, parts for vintage cars, parts for vintage watches, vinyl records, etc. but often before that happens the entire known supply needs to be scavenged.
So I can imagine a world where we manage to start doing the things we are doing with some new technology that is not digital silicon based computers that run on electricity. But there will be edges where the new technology is not compatible with some system, and in that edge lives this kind of speculation.
This story would be completely different if the old silicon computers were replaced with neo-computers.
Also for anything that uses a standard floppy drive there are cheap and plentiful adapters to use flash storage instead of disks.
The problem with those are that not all equipment that uses floppy drives conforms to standards. If you have a 50 year old embroidery machine from a company that no longer even exists you'd need someone pretty fucking technical to pull that shit apart and figure out if any of those drive replacements will even work in your weirdass machine. If you are lucky your machine used some off the shelf drive from a reputable company (I've seen some old weird drum machines machines that had Apple Disk ][ drives hanging off them) but otherwise...
I reckon the primary means of computing for the lay person becomes interacting with prompts to make the computer do what you want, but the actual knowledge of software engineering and computer science is gone. Even the prompts, due to linguistic evolution, are in a language that regular people don't understand and people's mastery of the language is akin to moderns trying to reconstruct something like Akkadian. We sort of know what specific words mean and the general grammar, but we simply don't know 90% of the vocabulary to express most ideas. Whatever the language model was populated on training data from an archaic language and is also full of weird idioms and phrases and misspellings that we can't reconstruct in an organized way.
In this context, the computers might be able to solve some basic problems for us, but everything we do with them is basically just received incantations from other crusty old people. Innovations keep happening, but they're small scale and mostly revolve around finding out new ways to tweak the incantations to do novel things. But the most we can do is say, like, "draw a picture" or "sort this list."