Is it a 7 foot height door? Is there an architectural style to take into consideration?

A disadvantage might be the higher price.

We have an 8 foot tall by 9 foot wide door with a curved arch at the top. We're thinking of replacing the damaged doors with a simple carriage house style steel insulated door.

In general (not a hard rule) more sections over the same height will mean a little more rigidity and a little more weight. Such a door will run smoother through a tighter curve at the top; this might be important with a low overhead and could be very important with an extremely low overhead. Some door dealers do not offer tighter radius track.

Regardless of residential track, a five section door will tend to travel smoother than a four section, and noticeably smoother than a three section door (I've never seen one of those!). Think of it as spreading those big panel flops across five smaller panel flops.

But with such a narrow door opening, just go for the looks.