Complexity of counting maximum number of co-linear points in Euclidean plane
First, my understanding is that the paper you linked to does not show that 3sum is incorrect. It just shows that the 3sum conjecture is false in a model of computation that is not realistic (i.e., we do not have currently a truly subquadratic algorithm for 3sum).

As for your question, if you are looking for a line containing more than $k$ points, and $k$ is sufficiently large (at least say $10\log^2 n$), then it is not hard to get a subquadratic number of candidate lines such that one of them with high probability is the desired line. Then using standard trade-offs between space and query time for range searching, you would get something subquadratic. The question is of course what to do in the in-between range, when the heaviest line has between $3$ and polylog n points on it. (For ref, see http://sarielhp.org/p/04/depth/depth.pdf). 

I think the interesting open question, is whether one can get $O(n^2/k^2)$ (the above would get you at best $O(n^2/k^2 * polylog n)$ [but its probably slower than that]).