[cygx]

there seems to be a misconception that a geodesic is a "generalized straight line". That is not remotely true. Geodesics, in mathematics, are "shortest paths".

Geodesics being generalized straight lines is exactly true. Also note that they are not necessarily shortest paths: In the framework of affine connections, they are defined as autoparallels.

Earth, for example, is travelling along a geodesic. But it is clearly accelerated towards the sun.

Earth is in free fall around the sun, so accelerometers will read 0. That's the whole point of General Relativity: Geodesic motion is not a consequence of Newton's second law, but the first one.