“As long as the sun stays put during the eight minute transit time everything works out.”
Therein lies the problem, the sun does not stay put but is acted upon by the mass of the planet, which causes it to ‘wobble’. That is, the sun and planet both orbit around a point which lies on a line joining their centroids, not at the sun’s centroid as would be the case if the sun staid put. Sure, the point is much closer to the sun than the planet due to the vast difference in their masses, but it’s not insignificant either (trying to avoid getting into the maths of it here). It is this angular change which would cause the gravity vector to shift off the perpendicular at the planet’s orbit and cause it to start losing angular momentum if gravity didn’t act instantly.
Hm, yeah, I see your point, and in a pure Newtonian universe the finite speed of gravity would cause a planet’s orbit to cave in over time like you say. But somehow relativity takes care of this problem. I’m not sure how, but I’m pretty sure it’s been accounted for. I know that’s a crummy reply but that’s as far as my knowledge goes at the moment.