First thing I would suggest from the application side of beekeeping is the real large plus of promoting bees that are not treated is you no longer have to perform this thankless and very much error prone process.  

A couple of snips... first from Jim Fischer....
'Not that it helps now, but counting varroa is best done with a sticky board
while the hive is still alive.  Over winter, sticky boards can also "map"
the size and approximate position of the cluster from the debris that fall
as well as allow one to screen for varroa that survived fall treatment or
came back to the hive with bees that robbed out weak colonies or fall
deadouts.'

And then from I think Charles Linder <hope I got the spellin' right..

'With all due respect,  no.'

'Best method bar none is Randys Wash methods.   Sticky boards are rudimentary
at best.  The comments that Mites don't cause winter losses is misleading
also.  Mites cause small winter clusters,  and super high death rate of
brood in that first round of spring.  You will also find mite feces right in
the cluster, no matter where it is.  It is much harder to spot in a honey
super than  brood nest,  but its there.'

From the academic side of my experiences I would suggest in almost all things there is no best.  The earliest side by side studies on the accuracy of varroa testing (at least those published in the ABJ) suggest to me that no matter which method you pick the error term is large.  As in most things that involve method and technique certainly the experience of the person doing the sampling is critical but even a large level of experience will not eliminate all the possible error term.  Accuracy and reliability are both compromised.  Certainly there is some plus to any method you PREFER and with this comes inevitable short comings.  We do a good deal of sugar shake testing at the TAMU Bee Lab and since we are set up there with screen bottom boards we also do sticky boards.  I would GUESS that alcohol wash could be faster.  I could likely see large benefit in the sticky boards in any location where you had a long winter spell in which any opening of the hives would be discourage.  I have heard from small scale 'hobby' beekeepers in the colder climate areas that do tell me they use the sticky board to monitor where and what is happening in the winter time by looking closely at the the debris path on the sticky board.... I am guessing this is what Jim Fischer has called mapping???  In somewhat the same fashion after the fact analysis (ie after a hive is dead) I look at the debris on my own solid bottom boards with a small magnifying glass to get some idea of varroa.  I commonly also look in the comb for varroa feces although certainly any robbing that may have taken place will mask this kind of evidence.  I look at both of these as equal valued bit of evidence for my own determination of cause.  As far as when death occurs I would also suggest over the long term (let say 50 years) a lot of hives die in February and March because the brooding rate of the hive and the available food resources relative to this brood rate are not managed properly.

Although with no treatment there is not much positive information provided by testing some of still do < as twisted as this may sound when I began to interact at the academic level to provide bees for studies in the current time frame I realized I needed to be rearing not only bees but also varroa mites.

Beyond the prior mentioned techniques I personally do a lot of casual testing for varroa by simply plucking capped drone cells.  

        

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