I used to draft wills for a living. One of the things I stressed to my clients was that making a will is a completely pointless thing to do unless the testator either tells someone where it is, or leaves it where it can't possibly be missed.
As your post implies, hundreds of thousands of wills get completely ignored because nobody knows of their existence.
Leaving a will with a solicitor is probably not a good idea. Firstly, this tends to happen when the testator has named the solicitor as one of his executors. That's hardly ever a good idea because the solicitor will charge a fortune to administer the estate, whereas (unless the estate is incredibly complicated) getting probate is incredibly easy to do. (No solicitor is required). Secondly, it makes far more sense to keep the will where it will definitely be found.
My own will is in the drawer where all of my other key documents (e.g. my birth certificate and passport) are kept. It's exactly where someone would expect to find it. It's in an envelope that has the word 'Will' printed in massive letters on both sides of it. Further, I've sent photocopies of my will to both of the executors, telling them exactly where the original is kept.
Chris