More generally, land value tax returns land to its rightful owner, the people in the community. No one created land so it doesn’t make sense to allow people to profit off it.

if that was the case, why does somebody currently own it, and who did this current owner buy it from (ala, go up the purchase chain back to the first buyer).

Eventually going up the chain brings us to the government, which again, had no right to give that land away when it belongs to the people. Whether they did it because of corruption or ineptitude is irrelevant.

So, let's take the scenario you gave: it depends very much where the mansion and the trailer are. Generally, people build mansions on high-value land and trailers are on low-value land, so the mansion will be taxed more than the trailer (a half acre in LA has a much higher value than a half-acre in rural Nevada where I live). If they are on the exact same lot, though, then yes they would pay the same tax. The idea is that taxing this way incentivizes the best use of land. Imagine that you have identical plots in New York, one is a big apartment complex, and the other a parking lot. With a property tax, you take a huge tax penalty for building, and so the housing complex pays a lot of taxes and the parking lot almost nothing. With a land value tax, however, the parking lot and the housing complex pay the same amount in tax, so it incentivizes the owner of the lot to build a more profitable housing complex there as well, and they take no tax penalty.

Prop 13 in CA is a law that freezes property tax assessments at sale, so you have people who are paying taxes on a home currently worth $2 million as if it were only worth the $150k they bought it for in 1980. This means it is always better just to rent the house out than sell it, since you have a huge tax benefit, making housing almost impossible to buy. Worse, you can pass the frozen assessment on in your will. So it is almost literally creating a quasi-landed gentry class.

This is absolutely not true. What prop 13 does is that it limits the yearly base property tax increase to 2%. It's not frozen, it still goes up every year, but only 2%. Note that this means it still goes up 2% even in years when the market is down. It's basically a damping fuction to absorb the wild fluctuations in market value.