The bit of fence in the foreground to the right of the gate suggests to me that the gate was there first: although if no-one involved in placing both the bridge and the gate had any common sense at all then both options are still open.
Yes indeed , or I just had a thought , maybe the bridge was there and a wooden stile had broken , with that fence being part of the original stile .
Maybe the policy being all stiles are replaced with steel gates eventually . But because the small tree is in the way and budget restrictions or conservation regs` etc...it ended up like a dogs dinner ...