(The title is mostly a joke, because Eckardt points really weren't the main difficulty.) Well, after another bunch of hours rewriting this proof in all aspects, I seem to have come to a final-so-far version of it: the statement is that weak approximation holds in places of good reduction for smooth cubic surfaces over the function field of a curve over an algebraically closed field. If that's Chinese to you, here's a slightly more understandable version of the same: consider a homogeneous cubic equation in four unknowns (say, X, Y, Z and T) whose coefficients are rational functions of one indeterminate (say, λ) having themselves complex coefficients: something like c3000(λ)·X³ + c2100(λ)·X²·Y + c2010(λ)·X²·Z + … + c0003(λ)·T³ = 0 where the cijkl are rational functions in λ with complex coefficients, and we are trying to solve for the variables X, Y, Z and T also as rational functions in λ; well, the idea is that if you fix a finite number of λ and for each one choose a solution of the corresponding equation (when specialized to that value), then there should be a parametric solution (one where X, Y, Z and T are rational functions in λ; I insist, rational) which interpolates them all, and actually, in a way I won't try to describe, one can fix the derivatives to any order of the variables X, Y, Z and T, not just their values at certain points; now actually this doesn't work for all λ: there are a certain finite number of them which can't be handled (places of “bad reduction”), essentially those in which the specialized equation becomes too degenerate in certain ways.
Now that I've finished that, I can perhaps start working on the piles and piles of emails that I still have to reply to.