Yes, the range gets quadrupled when the launch speed gets doubled.

The correct reasoning is already given with the question. Let us go over this reasoning once more, this time with the mathematical expressions in front of us.

The horizontal velocity component v_x is related to the speed v by

v_x = v cos q .

Similarly the vertical velocity component v_y,

v_y = v sin q .

These expressions show that both components are proportional to v and therefore get doubled when v gets doubled.

The vertical velocity component varies with time according to

v_y(t) = v_y(0) - gt

whence the time t to the top, where v_y(t) = 0, is given by

t = v_y(0)/g.

Thus, this time is proportional to the initial vertical velocity component which is proportional to the launch speed and therefore will double when the launch speed is doubled. Thus the total time of flight, which is equal to 2t, will double also.

Since the horizontal velocity component v_x(0), which is constant, is doubled and the time of flight is doubled, the range R, which is equal to the product of these two factors,

R = v_x(0)x2t,

is quadrupled.