we cannot assume that field lines start at some infinity and end at individual charges. In fact, lines start from one charge and travel to infinity and then come back to the other charge.
Since flux is defined as integrated electric field over a surface, you can enclose your charges with identical (Gaussian) surfaces and attempt to calculate the integral. By Gauss's law, however, this integral is proportional to the enclosed charge. Then, it is trivial to see why bigger charge contains more flux than the smaller charge.