G-d of course has no literal throne and doesn’t need a place to sit. What is the Midrash trying to tell us? Imagine you come to a house, look inside, and you see the place is bereft of furniture. Your first thought is that the place is abandoned. The President or the Queen of England may have just left that house a moment before you peeked in the door, but as far as you are concerned, the house is just an empty shell. Now imagine you come to a house and peek inside and there is a throne sitting right in the center of the room. First thought: this is a palace of some kind. A king must live here! No matter that there is nobody home at that moment – as far as you are concerned, this must be a special place.
The Ne’os Desheh (Ishbitz) explains that the Avos taught the world about G-d, but the world forgot the lesson just as soon as the Avos were out of sight. The Torah tells us that Yitzchak found that the Plishtim filled in and covered the wells which Avraham had dug just as soon as he was off the scene. Pharaoh forgot about Ya’akov Avinu and Yosef and their G-d as soon as they were gone. Out of sight, out of mind. When the king has no throne, as soon as he is out of the room, people just see an empty building – they forget that there is someone in charge. But when the king has a throne, even if the king is not present, that chair is a reminder of his presence. Yetiz’as Mitzrayim and the splitting of Yam Suf made such an impression even on the nations of the world that they could no longer forget that there was a G-d – even if they couldn’t see him, the events that happened were like the throne that stands in the middle of the room reminding everyone that this is the palace of the king, not an abandoned building.
With this idea in mind I think we can better understand the end of the parsha as well. Chazal comment on the words “ki yad al keis K-h” that the missing letter aleph in kisei, chair, indicates that Hashem’s throne is not complete so long as Amalek is present in the world. When Amalek runs wild, then people think the house is abandoned and they can do what they want.