(One more attempt at a simple and complete solution.)
Each chord must divide the circle into the ratio 4:3, so each chord must correspond to an arc of measure approximately 2.91624. Consider two such chords, one in a fixed location, and the other moving through all possible configurations in which it intersects the first chord. For simplicity, let the area of the circle be 7, so that the fixed chord divides the circle into regions with area 3 and 4.
(Note that if the chord were to continue rotating counterclockwise, what is currently the "bottom" of the chord would intersect the fixed chord, but any such configuration is the reflection image over the perpendicular bisector of the fixed chord of a configuration in the animation shown.)
As the chord rotates counterclockwise, the point of intersection of the two chords and the endpoint of the moving chord on the upper half of the circle both move solely leftward, so the area of the blue region is monotonically increasing and the area of the red region is monotonically decreasing as the chord rotates counterclockwise. The areas of the colored regions vary between 0 and 3, and are continuous functions of the rotation of the chord. There are 2 positions of the rotating chord for which one of the two colored regions has area 1:
In each configuration, we can determine the areas of the regions by symmetry (exchange the moving chord and the fixed chord). In the left diagram, the areas are (counterclockwise, starting with the blue region) 1:2:1:3. In the right diagram, the areas are 2:1:2:2. Since the final chord must divide 3 of the existing 4 regions in order to have 7 regions, the configuration with areas 1:2:1:3 (above left) won't work, since dividing either region with area 1 won't give us 7 regions with area 1.
Now, working with the configuration with areas 2:1:2:2 (above right), add a third chord, rotating through all possible positions that divide the lower right region.
As before, the area of the purple region is monotone and continuous as the chord rotates and is between 0 and 2, so there is a unique configuration where the area of the purple region is 1.
Considering the position of the purple chord relative to the original fixed chord, the small wedge region at the right cannot have area 1, and this can be verified computationally. (It is also the case that the area of the triangle in the center cannot have are 1 in this configuration.) So, it is not possible to divide the circle into 7 regions of equal area using 3 chords.