I think a few things happen that can cause even people who already own homes to get effectively priced out:
1. Through life circumstances, they need to move. The most common case is when they have kids and need a bigger home, but sometimes they need/want to move closer to a new job elsewhere in the city, neighborhoods change in ways they don't like, kids move out and they want to downsize, etc.
2. Property taxes do go up, yes.
3. When real estate goes up, everything else goes up too. People that can afford more expensive homes also generally have more disposable income. So they influence prices in the market and food, restaurants, services, daycare, etc. all get more expensive.
> And can start a new life in a cheaper place - probably getting about same money for the same job - but with whole lot of money in the pocket.
This is a thing, yes. You see older people move away from the city once their kids move out. When your kids are young, it's good to be in a city and stay put so that they can get a good stable school, long-term friends, and access to lots of learning experiences. Once the kids have left, though, that all matters less.
But there is a cost to that. Uprooting and moving means leaving behind friends and familiar places. Diminished friendship among the elderly is one of the causes of suicide and lower life expectancy. It may mean moving farther from their own children.
For example, I live in Seattle. My mother-in-law used to be a twenty minute drive. We spent a lot of time with her and she spent lots of time with her grandkids. Every year, the rent went up on her apartment until she couldn't afford it. She moved out to the country. It's a beautiful house, but it's an hour away, so we see her much less often, and that's a real loss to all of us.
> Why should we feel sorry for them?
I think that's a shallow way to look at it. We should empathize with them, with both the good and bad parts of their experience.
Overall, I think it probably is a net win to have cities be mostly oriented towards professionals and parents and that it helps balance the demographic distribution of people in the US to have older people move to rural areas. But there are downsides too.