In Hnefatafl (with these rules) the aim of the white player is to move their king to one of the corner squares. How do I know when it's safe to move the king? Are there 'signs', for example, the other player not having many pieces, or those pieces to mostly be on one side of the board?
In my experience (albeit with a slightly different tafl variant), a key part of white's strategy is maintaining the ability to play towards any of the corner squares.
The throne square isn't intrinsically all that much safer than any other square on the board. Initially, keeping your king near the center makes it easier to take advantage of momentary gaps in unexpected places. Moving your king is mostly about maintaining this sort of flexibility -- if there are a lot of enmeshed pieces near the center, for instance, running your king off to one side might open up the ability to both play towards the two nearest corners and cut around the jam to get to one of the far ones.