As even that top article points out we'd expect the rate of unvaccinated people to approach the base rate which is 76% of the population having received one dose. And since old people are at more risk and are more vaccinated, we'd expect it to reach even higher eventually. But that happens because eventually the unvaccinated do all pick up natural immunity and the effect we're measuring is that the excess load caused by the unvaccinated is declining. We could have gotten to this point much quicker by having 100% vaccination rates (and 100% of the people still being admitted with COVID would then be vaccinated) but with an order of magnitude less load on the hospital system.

The top article you cite has all this information in it, and even mentions the base rate fallacy. It is criticizing the framing of the pandemic as a "pandemic of the unvaccinated" and trying to focus attention on the elderly, but that is orthogonal to the question of if the unvaccinated are disproportionately clogging up the hospital system. Both statistics are true. Everyone should still get vaccinated to reduce the load on the hospital system (although this concern is now fading as antivaxxers actually do pick up immunity the hard way). We should also focus on the elderly more.

As a simple example: if a population begins entirely susceptible and there's a vaccine from day 1 which is available and reduces the risks of hospitalization by 9x and 90% of the people get the vaccine, then the rates of hospitalization will be split 50/50 between the vaccinated and unvaccinated. Vaccinating the rest of the population would increase vaccination rates by 10% but would decrease overall hospitalization rates by 44%.