Here is another example.:
Two planes are going to fly from Love Field in Dallas to Meacham Field in Fort Worth, about 30 miles due west.
One plane goes due east, the other plane goes due west.
How far would the plane going east have to go and how far west would the other plane have to go ?.
Which plane would get to Meacham Field first , considering both planes fly at the same speed ?
Would only one plane make it to Meacham Field according to the FE Map if you could just fly in a straight line on the FE map ?
And how would this work on the Round Earth if you just flew in a straight line on the Round Earth ?
As far as I've seen here, no accurate map of the flat earth exists, nor a map of any kind beyond a tentative doodle or a projection of a globe. That said, in
principle I think the claim would be that the westbound plane would reach its destination the same as normal and the eastbound plane would do one of two things depending on how you interpret your question. East on the flat earth is just a big circle centred on the north pole so kept a constant easterly bearing by turning north as it flew its path would be a big circle around the earth roughly along the flat earth equivalent of the 30th parallel. However, you also specified a straight line, so if it set off eastward but carried on flying in a straight line it would head roughly toward the Atlantic somewhere, maybe cross some land but eventually hit the edge of the world and not take any pictures or come back again.

Flying in a straight line of any bearing on a globe takes you in a great circle and you end up back where you started, excluding the rotation that's happened during the journey.
/edit, straight courses being great circles on a globe also answers Theo's rephrased question about leaving australia in two directions and getting back there without changing course. If you fly straight for 40,000km in any direction, you end up back where you started. So, yes if two planes left in any two directions you can name and flew straight, they'd get back home eventually without having to change course.