[bojan]

Why don't you use old record player as rotating plate?
I was experimenting with such "out-of-the-square-box"ideas a lot in the past.
Including chamber with lower air pressure behind the thin membrane (mylar).. or high electrical field even.. concept always worked, but that was all.
At the end, you will find that there is a reason why glass is used for mirrors... It is because only glass can keep the shape within required tolerances (1/10 of the wavelength) over wide temperature range etc, while it is reasonably easy to figure it precisely enough [speculum (used in the past for mirrors) is another example of good material... but obviously not as good as glass, otherwise it would have been used already]
Even with glass, if the mirror is not mounted properly, things can go wrong (in optical sense - mirror pinching is notorious, for example.. temperature gradients inside the glass are more subtle but still significant).
This method may be useful though, to obtain the initial shape of the glass for grinding - polishing. I imagine it is used for manufacturing large mirror blanks....