In the longer term property taxes reduce house prices.
No, in the real world when you put a tax on something its price increases.
They also encourage empty-nesters to move out of family homes which can then be occupied by families.
There is nothing to stop families buying such properties now, they just haven't offered a high enough price to entice the owner to leave. In any event, in a free society, the government has no business deciding that certain categories of citizens are more deserving of outcomes than others. If you want an outcome, e.g. a certain type of house or a certain make of car or go on foreign holidays, you pay for it or go without.
The social infrastructure, particularly schools, doesn't have to be duplicated around the country as often etc.
This is just soviet style planning. In today's world, you live where you like, i.e. can afford to, and then travel to work; to school; to shopping, entertainment, etc. of your choice.
When their private pension funds would have been wiped out after the financial crisis their children and grandchildren bailed them out.
Actually, this never happened. The government introduced a 0.6% levy that was applied to the full value of funds in private pensions schemes that netted, i.e. confiscated, over 2 billion for the exchequer. There was and still is a pension levy on public sector pensions. So pensions, both private and public, bailed the state out, not the other way round.
If they bought their house during a period of high taxed and high interest rates then they also bought during a period of very low prices and high inflation.
In the real world, 'Low prices' and 'high inflation' are mutually excusive. High inflation is after all a secular increase in price levels. Houses are by and large priced to represent a multiple of implied rental earnings or a ratio of price to average income. Houses were no more 'cheaper' in the past in relative terms and were priced in terms of lower wages, higher unemployment, and higher interest rates. Remember, you don't actually buy a house. You buy a mortgage and swap it for a house. Interest rates in the past were up to 17%. Today's mortgage rate of 2.95% is as close to free money as you can get, even in a low inflation environment.
Their children and grandchildren could have their own big garden to play in every day if more older people downsized.
This is just jealousy. You are basically saying that someone else's children should enjoy a garden but not the grandchildren of the people who own it. You can't rationally decide between the marginal happiness one set of children get from playing in a garden and another set.