Why do we need array antennas?

Long wire antennas can do that, but only in two of 3 axis.

And the usual narrow band response of a dipole has been ruined.

Array antennas are the basis for beam forming. With array antennas, you can control the directivity by driving each antenna of the array with a different phase.

If we can increase the directivity of the single element antenna by increasing its electrical size then why do we need array antennas?

Increasing the size of a single element means it will certainly receive unwanted lower frequencies and this might make the electronics more complex.

It also means that the impedance of the antenna at the "wanted" frequency changes from the straightforward near-resistive impedance of the half-wave dipole antenna (for instance) to a less useful higher impedance as shown below: -

So, when the dipole is dealing with a frequency corresponding to half a wavelength, it's impedance is around 74 ohms resistive but, if the length doubled the impedance sky-rockets and this becomes a bigger problem to receive.