Argument A: reciprocity
By reciprocity, we can flip this question around and ask it from the perspective of the transmitter. Assuming we've done the obvious like minimize feedline losses, and we're holding the geography, frequency, and so on constant, how can we get a higher fraction of the transmitter's power to appear at the receive antenna's feedpoint?
If the transmitter has some omnidirectional antenna like a vertical monopole, most of the radiated energy is going in the wrong direction to ever reach the receiver, and is thus lost.
Increasing the directivity of the transmitting antenna sends a higher proportion of the transmitter's power towards the receiving station, accomplishing the goal.
It should be intuitively obvious from the law of conservation of energy that a higher gain is the only option here, which if losses are already minimized means a higher directivity.
Argument B: phase coherence
Consider you have an omnidirectional antenna like a vertical monopole, and the received power at the feedpoint is 1μW.
Wanting to receive more power, you install another identical antenna beside it. This 2nd antenna, having a nearly identical path to the transmitter as the 1st, receives another 1μW.
In total you're receiving 2μW, but to get the full 2μW combined into a single port requires that the phases of the two antennas are adjusted so they add coherently.
Unfortunately, this means the array has acquired more directivity. While some phasing arrangement may work for one transmitter location, there will be other possible geometries which place the transmitter closer to one of the antennas and farther from the other, thus altering the phase at which the signals combine. Thus the antenna array is no longer omnidirectional.
So, while a longer wire, or a larger array of antennas will indeed intercept more power, it's not possible to coherently combine those powers without increasing directivity.
An array of antennas does indeed receive more power, the difficulty is it's often unknown what phasing of the antennas will result in the signal combining coherently. One solution is to make the phasing variable, and to dynamically adjust it for maximum signal clarity.
Modern communication systems (Wi-Fi, cell phones) can do this dynamically by separately digitizing the signal from each antenna, then dynamically adjusting the phasing in software. This is effectively an antenna array that's always "pointed" in the optimal direction, thus obtaining a higher link quality without the disadvantages of a directional antenna.
For terrestrial communications in predominately flat terrain, it's safe to assume the other station is near the horizon, that is, not flying overhead, and not underground, although the direction may be unknown. In these cases it's possible to maintain a omnidirectional (meaning uniform along the azimuth) pattern by increasing the directivity in the elevation axis. That is, making a "flatter" doughnut.
Often this is achieved with a collinear array. A quarter-wave monopole above a ground plane has gain of 5.2 dBi, and a dipole 2.2 dBi: when you see omnidirectional antennas with higher gains they are usually collinear arrays with a "flatter" pattern.