Hot take: Delphine is supposed to be insufferable when she gives you the horn of Juergen Windcaller

Her reaction is designed to emotionally push the Dragonborn to actually leave. It is a variant on Caius Cosades in Morrowind when he tells you to go make something of yourself before continuing the main quest. The Dragonborn's response should be "I don't have time for this" and then go adventure for a bit. Random dragon's won't spawn and it will be easier to do guild/random quests.

Her dialogue when you return to her the 2nd time is more gentle.