The sun's path IS curved though, just, somewhat like the horizon viewed from altitude, not a lot. Ignoring the horizon, if you just take a timelapse of the sun's path with a rectilinear camera (like, roughly, an iPhone) then it's going to be slightly curved in that image (unless it's on an equinox)
If you do a cylindrical projection where the axis of the cylinder is aligned with the globe axis, it ought to be straight (and the horizon ought to be bent). Basically, use a pinhole camera with a semicircular projection surface and put the pinhole in the center of the diameter. I think what happens is the horizon gets twisted for our usual perspective (unless you project onto a half-sphere and not a half-cylinder). ;-p
I followed a youtuber a while ago who tried to verify that the lit part of the moon always points at sun, and with a half-moon, in the evening the lit part of the moon would still point up while the sun was lower in the sky that the moon. If you swivel the camera, or take a panoramic image, the moon doesn't seem to point at the sun! He didn't give up, and eventually used a stick to verify that the moon was pointing at the sun if you followed the direction across the "dome". On the panoramic photograph, you'd have had to draw a curve, but in reality, the connection was straight.
If you draw two great circles on a sphere, they're both straight, but their relative angles are shifting along their length. This is what happens here, and it happens at the equinox, too. (If the sun rises and sets, and the horizon is straight in our frame of reference, then the path of the sun must change angles. Except the horizon isn't even a great circle.)
For 12 hours, the sun curves ("appears to curve") one way, and the other 12 hours, it curves back. This is why, if the sunrise/sunset occurs near 6am or 6pm local solar time, the sun path curvature against the horizon is the least. This is also why the moon appears to rotate (it doesn't if you use an equatorial mount).