Before it was widely accepted that the Earth was round, how did astronomers describe the yearly movement of the sun to south in the Northern Winter that produced the seasons? Did the Earth wobble or did the Sun just not have a perfectly circular trajectory around the Earth? Did this perspective change when we realized the Earth was a globe? What about before Heliocentrism was the norm?
This is not really a physics or astronomy question, but I reckon that when you model the Earth as flat, the stars and the Sun are merely minor objects and their motion is governed by heavenly spirits rather than regular orbits. So, rather then the flat Earth tilting, the Sun's trajectory changed with the seasons.
Btw, what I find more concerning is how the (infinitely extended) flat Earth can be reconciled with the Sun rising in the East after setting in the West: how did it get there?
Btw2, I don't think that elites in classic times (Greek/Egyptian/Middle East Cultures ~1000Bc and more recent) seriously considered the Earth to be flat. All the constructions of epicyles etc for explaining the data assume a spherical Earth in the centre of the cosmos. Flat is much more arcane than Geocentric.