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I studied in my physics textbook that a plane mirror is basically a glass plate a few millimetres thick. The upper surface of the glass plate is highly polished and smooth while the lower surface is painted with silver. It struck me that we could have just simply utilized an opaque surface painted with silver at the top. Why do we need an upper smooth, transparent surface?

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You are right. And that is how high quality mirrors for telescopes are made.

But putting the mirror coating on the top surface leaves it exposed and easy to damage. You ouldn't want your bathroom mirror to be like that.

It is cheap and easy to make flat, smooth glass. It is easy to deposit silver on the back and protect it with a coating.

If you have a top surface, you need it to be flat and smooth so it won't distort the image.

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The mirrors we use in everyday life put the metallic coating at the back to protect it from getting scratched or abraded. So it has nothing to do with optical performance and everything to do with practical considerations. The top surface of the glass has to be smooth because if it was rough it would scatter light and make the mirror look dull.

High performance mirrors do indeed put the metallic coating on the surface as you suggest. For example the mirror used in the Hubble space telescope has a glass base with a layer of aluminium on the top. This works fine because in space there is nothing to corrode the aluminium and of course the mirror is never going to be touched so it can't be scratched or abraded.

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    $\begingroup$ a mirror with the reflective coating on top is called a front-surface mirror. its optical performance is superior to that of a back-surface mirror because the back surface mirror exhibits a slight amount of reflection at the front surface as well as the main reflection off the back, causing the resulting reflection to contain two images that do not line up perfectly. Since the most common mirror metallization is aluminum and since it is subject to slow chemical degradation, the front-surface mirrors used in big telescopes must be resurfaced with fresh aluminum every few years. $\endgroup$
    – niels nielsen
    Nov 13, 2017 at 8:59

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