In the U.S., you as a landowner can be sued if someone gets injured on your property while trespassing. Understandably, people close off their land.

Same in the UK - it is called the Occupiers Liability Act. It's not black and white but the liability is on the Occupier. I know this because I own an area of ancient semi-natural woodland which I'm restoring back to the natural NVC type (basically, oak standards and hazel coppice). As part of my duty I have to go and survey all the boundary trees once every "reasonable period" and after storms or significant weather events. There's some nuance to in the liability I've got when considering the harm to trespassers, but staying on top of duties and having your auditing ducks in a row helps.