You are not the only responder to have stated that a mono mix should be made
before de-clicking and other noise removal work is done.  I have heard this view stated in the past, but I can't agree.

I find the best contemporary digital de-clickers are so good that they fully remove the distortions caused by scratches. Once these and other defects are replaced by sound synthesized from surrounding audio, they effectively vanish and do not depend on mixing for removal. 

As you have mentioned, the stereo transfer of a lateral recording allows distorted areas on only one groove wall to be manually replaced by a less distorted section from the corresponding section from the opposite wall, and this is indeed a powerful tool.

A mono mix prior to de-clicking just combines the non-vertical noise from both channels. I find processing the stereo before making the mono to be a superior approach, but I understand there are those who disagree.

Because the stereo cartridge responds to motion around a 360 degree arc, stylus motion will ordinarily contain both lateral and vertical component.
This is the reason why mono mixes should always be made very carefully, and I always urge monitoring the difference between the two channels when making the mix because our ears are far better at tuning a perfect null than a perfect sum!  (It is not difficult to crete a sum-and-difference matrix using a balanced audio mixer which has a polarity switch on at least one input - which will probably be mis-labelled, "phase".)  

And yes, the mid-side microphone technique matrixes a mono component (M) and a side component (a signal split in two and out of polarity), and it is, in effect, a sum and difference mixer. 

Other factors involved are, as Jamie Howarth noted, disc cutting tools were not always absolutely perpendicular to the wax or lacquer surface.  Also there's a phase shift due to the L & R arrival-time differences which accompany the playing of a lateral groove with a radial arm, which is perfectly tangent to the groove at only two places on a disc side, and the "azimuth" correctors in Cedar and iZotope can correct this.

Thoughts on vertical recording: it's not perfect, as Oliver Berliner will quickly tell you if you let him.

Pierce and Hunt's paper (Journal of SMPE, 1938) identifies the distortion produced in vertical recordings by the difference in the shape of the disc cutter and the playback stylus, which they called the "poid", this is vertical tracing distortion. Lateral recording also suffers from a form of this.

Worse, vertical recording requires that the groove depth varies with the modulation.  The problem with this is that as the cut gets deeper, the mechanical impeding or loading of the recording medium to the cutting stylus gets higher.  This is a nonlinear loading phenomenon, by definition, a classic cause of asymmetrical distortion which produces even ordered harmonic distortion products. 

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> Date:    Wed, 6 May 2015 10:42:51 -0400
> From:    John Haley <[log in to unmask]>
> Subject: Re: Playing Edison Diamond discs
> For myself, I do not "Declick" or run any noise reduction before combining
> the channels (which I do on the computer, not from cartridge wiring).  The
> reason is that you want as much noise as possible to cancel when combining
> the channels. The noise that is going to cancel is in both channels but
> with opposite phase.  For example, although a loud click may be louder in
> one channel that the other, the less loud one will still be the
> corresponding "negative" that will bring down the louder one in other
> channel, even if not completely.  And a number of such "pairs" are equal in
> strength, or close to it.  Anything you have done to one channel or the
> other with a computer program will interfere with this cancellation
> process, so combining the channels should be done first.  The same is true
> for ordinary groove wall noise in a mono LP, much of which will cancel
> itself out when the channels are combined.
> The reason not to wire the cartridge to mono is that sometimes one channel
> (groove wall) is a mess while the other one comes thru OK, and you want to
> be able to access the good side to assemble to best result.  If you have
> already combined the channels, you have put the mess in both channels
> already, where you can't pull the channels apart.  So I keep an original
> "stereo" version on the computer, even though I am working on the one where
> the channels have been combined, in case I find that I need it.
> Best,
> John Haley