Obviously, I'm not an official of PRoC so I can only reference official statements and provide speculation from their point of view. It's rather difficult of course to find any sources that don't immediately look like propaganda so a lot of this will be speculation for what can be logical reasons.
First let's point out that relative to the countries that you are comparing with Chine in the OP, China has a low median age of 38.4 years. Similarly, it has low elderly dependency ratios. Compared with the rest of the world China has a low urbanization rate and furthermore (this you'll have to do some data mining for from the previous source) the urban rate is a skewed towards the younger population. That means that the elderly are relatively few and in remote regions. It's far harder to obtain vaccinations than in the city centers and the numbers many not be worth it due to the low impact of spread in remote villages and the low opportunity of spread to a remote region due to lack of traffic.
Had to use US data here due to lack of Chinese sources, but it shouldn't differ worldwide unless the CDC is purposefully misrepresenting death stats. As you are doubtless already aware, death rates are higher for people with weaker immune systems (as is the case with every disease) such as elderly people. Combining with the previous point on logistics, it may not be worth the effort to address this demographic as they are high risk regardless. People with weaker immune systems (such as elderly people) tend to die quickly so have a lower opportunity to spread the disease. This can bee seen as a reasonable triage situation; if your goal is only to stop the spread of the disease, not to prevent people from getting it, then you will focus on the elderly in urban areas where they can spread it. Since the elderly are skewed towards rural areas, this means that they will have an appropriately lower proportion of vaccination.
As expected from a former socialist country, China does have a somewhat functioning universal healthcare system (with caveats on about every word there...). As elderly care is generally more frequent than the younger demographic, it may be in China's best interest to simply let these people die than to perform expensive treatments for long periods of time trying to keep them alive when they can no longer work to support the system at a later date. Even if China wasn't an authoritarian state™, if I were in charge of a country and my main focus was economic growth to compete with the US, I would see myself sacrificing the retired members of my country. Health care aside, funds like pensions are a drain to the state only incurred from retirees. Not to mention, when these systems were put into place, the math may have made sense (work for 40 years from 25 to 65, get 10 years of about 1/4 of your wage) but falls apart quickly when people are retired for 30 years due to increased life expectancy and turns into a Ponzi scheme. Hence, there may be economic motivation to have the elderly die off from the virus but to save the younger, working population.
I do want to point out that given the size of China (largest in the world by hundreds of millions of people) and the urbanization rate of China (relatively low), such a high vaccination rate takes a large amount of effort that doesn't scale linearly with size. Considering the massive effort already undertaken, this is likely a deliberate decision so the question
So why didn't China use their powers to force 100% of the elderly to get vaccinated?
Is based on a false premise that they want to do so or it is even in their interests to do so. In general questions like "why doesn't X use their powers to do Y" are flawed because they assume in the first place that X wants to do Y. Not to mention, the inquiry in question asked about a 100% vaccination rate which is likely an exercise in futility particularly for China. Not even taking into account the rural/urban rate the total population is extremely uncertain. The level of uncertainty between the max and minimum estimates of China is on the order of hundreds of thousands to millions (a quick look at the demographics page on Wikipedia shows 3 different numbers for total population throughout the article), so likely their own government has a bit of trouble accurately determining the size of their population so 100% anything is out of question.