Gravitational redshift paradox

How can starlight appear gravitationally redshifted by the Sun, if their light is blueshifted as it falls all the way through the sun's gravitational well, and is then redshifted by a smaller amount as it climbs a much smaller distance to Earth ?

 When observing a star close to the sun from Earth’s position, the observed shift should actually be a blueshift, not a redshift, because the starlight travels a much longer distance falling into the sun’s gravitational well than climbing out from it as it reaches our telescopes on Earth (or its orbit) ! So the blueshift from gravitational potential should be much higher than the redshift. The fact that we see a redshift means that it is 100% NOT a gravitational redshift, but a plasma redshift from energy loss in the sun’s corona.

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