You have to control vaccination rates and hospitalization rates by age. Younger people are less vaccinated and at lower risk of vaccines which gives a positive correlation between vaccination and hospitalization, even though for every age, the risk of COVID hospitalization is lower if you are vaccinated.

We get this (sorry for the bad formatting):

Hospital count  Hospital %  Vaccine Rate %  # cases unvaxed if unvaxed did nothing  

< 12 9 10.20% 35% 5.85
12 to 19 0 0% 88% 0
20 to 39 1 1.10% 90% 0.1
40 to 59 7 8.00% 95% 0.35
60 to 79 41 46.60% 100% 0
80 and over 30 34.10% 100% 0

So if the vaccine did nothing we'd expect 6.3 unvaccinated cases in total. Instead there is 17. So there are more unvaccinated people hospitalized than we would statistically expect. The fact that there would be any elderly unvaccinated in the hospital shows how much it matters. <1% of the elderly are unvaccinated there and yet I wouldn't be surprised if they made up some of the elderly hospitalized people.

Data sources:

https://www.princeedwardisland.ca/en/information/health-and-wellness/covid-19-testing-and-case-data

https://health-infobase.canada.ca/covid-19/vaccination-coverage/ (PEI data)

Age prop_atleast1dose

0–4 0
05–11 70.07
12–17 88.87
18–29 82.31
30–39 98
40–49 94.51
50–59 96.46
60–69 >=99
70–79 >=99
80+ >=99