Isn't that a clear indication that there's something besides "general trends in old people" going on here?
Looking at the tables, you might get the idea that COVID hospitalization causes a lot of diabetes. The way they arrive at that is to count the diabetes diagnoses made after COVID admission.
However to me it seems far more likely that a diabetic without a diagnosis (which is common) gets hospitalized with COVID, then gets their diabetes diagnosis in the ensuing medical surveillance. Having an undiagnosed liver or heart issue is also common.
(There are other good questions to ask too, of course.)
Why deviate?
They do justify their choice in the actual paper (e.g. see the end of the "Study population" section).
(It's also interesting to compare COVID hospitalizations to other ones, say from a heart attack or allergic reaction, but those are different questions. For example, maybe a COVID hospitalization is less bad than one due to a heart attack, or worse than an allergy. Or maybe any hospital visit is just bad period. But given many people got hospitalized by COVID this year - something that never happened in the past - we need to know what that event cost them, that's the point of this study.)
How do you then control for those with a chronic condition that is undiagnosed? If somebody is hospitalized, they'll perform all sorts of tests on them, that's how many people get their diabetes diagnosis in the first place.
Looking at the charts, you might get the idea that there's a staggering increase in new-onset diabetes following COVID infection, which doesn't seem plausible to me. Yet, right next to that, you see damage in heart, liver and kidneys, and this gets picked up uncritically as an outcome of COVID. The hypothesis that COVID causes organ damage is already well-established, so this just confirms what we expect. What we don't expect, we just ignore, even though it might point to a flaw in our methodology.
But we know that COVID admittances caused a big increase in the total number of people hospitalized - to the point of overwhelming some health care systems. That strongly suggests that underlying conditions would not have sent as many people to the hospital anyhow if COVID had never appeared and infected them.
It is possible you explain the mechanism, though. Perhaps there are lots of people with undiagnosed underlying conditions, and getting a serious case of COVID is enough to worsen those conditions into life-threatening ones. (Perhaps this only hastens the inevitable for some of them.) If this is the mechanism then both you and the article's hypotheses would be correct.