As of yesterday it should now do this!
I kind of already did this, you could click the "Lemmy.world" in the header in the top left and it'd refresh the feed, BUT only if your page URL would change. If you were already on https://alexandrite.app/lemmy.world and clicked the header, because that's the same URL clicking the header would go to it just wouldn't do anything. However if you had changed some sorts so you were on https://alexandrite.app/lemmy.world?type=Posts&listing=Local&sort=Hot or something, clicking the header text would refresh the feed.
Now it'll just always refresh the feed, though it's a little slower because it's reloading extra data it doesn't need to, but I'll probably optimize it a bit sometime in the future.