(I have some experience with telescope mirror making.)
It would not work as described. It's not even close.
In a reflector telescope, the primary mirror needs to satisfy the Rayleigh criterion, which states that the surface error must not exceed λ/4. For visible light, that means 100 nanometers.
The ideal shape of the primary mirror in regular newtonian scopes is a paraboloid, so no matter what technique you use to make the mirror, the resulting surface needs to follow the ideal parabola with an error no greater than 100nm (due to Rayleigh).
For a mirror of reasonable aperture and focal length, your flat segments would diverge from the parabolic surface by an error far greater than 100nm, more like fractions of millimeter or so, which is about 1000x bigger than the acceptable value.
There's just no way around it. Each part of the mirror needs to track the ideal parabola with no more than 100nm error. Otherwise the mirror is not usable for visible light.
The technique you describe would be usable for microwaves. Since the wavelength in that case is on the order of 1cm or more, λ/4 would be a few millimeters. In that case, a wide shallow parabolic bowl, having multiple flat metal disks glued on the inside might work well enough. If the size of each metal disk is comparable to the wavelength, then it would definitely work well. Just make sure the error in each point is no greater than λ/4.
(Okay, you would then get some resonance effects in each disk itself, which would change the overall result, but let's not go there. Bottom line: it's feasible with microwaves, not feasible with visible light.)
If you don't need to create an image at focus, but just collect the energy of a microwave beam, then the error could be bigger than λ/4; the only ill effect would be that your reflector would become less and less efficient as the error increases.
One more thing: In theory, a mirror with an overall surface error of exactly λ/4 could have a total efficiency of exactly zero, but that's a pretty pathologic (unrealistic) case. In most real life situations, a few λ/4 zones here and there simply reduce the overall energy-gathering power. So keep the error down and it will work well.