You could similarly run the police at a profit instead of being a cost center by gifting 100 hectares around every police station to the police, and have them collect rent on it. That doesn't make sense either.
For one, it creates a perverse incentive where the government can't let property prices fall for the common good least their public transport (or police) services would go out of business.
Hong Kong famously has a problem with this. It's a relatively low-tax regime with absurdly high housing prices, because they raise a much larger part of their revenue by taxing land than most other similarly developed areas.
Not if you tax the value of ground rent only. If anything, the effect goes the other way, as owners are incented to put their underused properties on the market. (And yes, taxing ground rent only in some relative small areas e.g. around a station may introduce distortionary incentives. But in general, shifting taxation towards land improves the incentive for good government, especially provision of local amenities.)
There is also strong synergy between the trains and the area immediately around it. Also you have to buy the land anyways for the station.
Don’t think that argument holds as strongly to police statins