Say there a 1 in 10 chance that you will end up in hospital with covid, it's probably something like that. Unlesd you have a similar base rate chance of being hospitalised the proportion of the hospital population with covid will be higher than the population outside.
Not because you're more likely to get it in hospital but because you're more likely to be in hospital if you've got it. If the over 65s make up roughly 1/4 of the population and 80% of admissions, they could be 4 times likelier to have covid at any given time and still not be any likelier than a 19 year old to have it while in the private residence population.
Your last paragraph may well be right but it isn't demonstrated by the report you quoted.