How could a 20 inch space telescope "be able to make out Earth-size planets" orbiting Alpha Centauri?
I'm not familiar with the design of the ProjectBlue telescope, but I think you have answered your own question.

The habitable zones for Alpha Cen A and B, are approximately centred at 1.25au and 0.7au. Both are at a distance of 4.37 light years.

1au at 4.37 light years, subtends an angle of 0.74 arcseconds. If working at blue wavelengths (the aim appears to be to find a pale blue dot, so let's say 400nm), an Airy disk of 0.74 arcsec is achieved with a telescope with a diameter of just 13cm. To stand a chance of resolving planets in the habitable zone, you would need a telescope a little bigger than this.

However, a bigger problem is separating the signal from the star and planets when they are barely resolved. The Airy disc is just the first minimum in the diffraction pattern. If the star is around 10 billion times as bright as the planet (ball park figure) then it would still swamp a planet that was just resolved.

Thus you need to build a bigger telescope to increase the angular resolution. And you need to build superb optics and a coronagraph to stop the light from the star swamping your detector. But this costs more money, the telescope has more mass, needs a bigger launch vehicle etc. So I guess a 50cm telescope is their compromise.

Is it achievable? I don't know. The best telescopes on Earth achieve almost diffraction-limited angular resolution using adaptive optics. However the contrast they can obtain for exoplanet imaging is compromised by the atmosphere to some extent. If you are looking for a "pale blue dot", then this can't be done from the ground because adaptive optics systems do not currently work at such wavelengths.

It is quite possible that Earth-like planets in the habitable zone of Alpha Cen could be discovered from the ground in the next few years using improved radial velocity measurements, but not by imaging.

An interesting question is why this can't be done now with HST? The proposed telescope will spend two years doing this. Even if HST could cut this to months, I am not convinced that detecting a faint smudge around the nearest star has the necessary science pay off to get the time, especially as there is a fair chance of failure. There may not be planets in the habitable zone or there may be sufficient zodiacal dust in the system to make the measurements impossible due to scattered light.

Further research suggests that the coronagraph on the HST ACS camera would only give around 7-8 magnitudes (factors of a thousand) contrast at angular separations of an arcsecond. So nowhere near what is required. This new telescope must have radically improved optics and a more advanced coronagraph.