During the quest Crisis at Hyrule Castle in Tears of the Kingdom, you will get a series of quest objectives that will have you travelling up the road to Hyrule Castle, needing to speak to different people along the way. One of the objectives will have you needing to speak to someone at the first gatehouse but speaking to him does not update the quest objective.
This is one of those unfortunate placements that should have been caught during QA but is not actually a bug, it is just a misleading placement of the objective pin. The pin on the map shows the guy standing next to the fire in the main gatehouse. He will talk about the rusted weapons but he is not the guy you are supposed to speak with. The video guide below will show you where you need to go.
The person you need to speak with is actually on the upper level of the gatehouse. You can actually get up here a lot easier by using the ability to rise up through the floor. I was just getting started with the game at the time and the idea didn’t occur to me but I guess there are several ways to get up there.
Speaking with the correct person who is standing on a ledge of the gatehouse, looking at Hyrule Castle which is now floating above the chasm is the correct person to speak with and once you have spoken to him, you should be all good with the rest of the quest.