There's a lot there and you should know that this is strictly an experiment (at least for me!) Others use squares routinely in 20th century gaming (PBI from Peter Pig) so this is something I've been merely toying with for awhile now but others have adopted enthusiastically.
When I refer to "certainty" I simply mean that when playing with very tiny lightweight playing pieces they will be more prone to shift due to irregularities in the table top terrain. It's the milimetrics problem seen in games such as DBM. PLaying with easy going gamers solves a lot of this problem but, like I wrote previously, even the most honorable gamer is going to have a tough time figuring out firing angles based on the tiny front of a 1/285 AFV (eg the Stryker is only 8mm in the very front!) It's simply that such tiny vehicles and figure stands can be very "fiddly" which presents an ergonomic challenge at times (especially as we get older and perhaps less patient!).
As for the geometry, you're absolutely correct so it's a trade-off. PBI has an approach that I've modified and adopted to mitigate the warping and it's this: the first two diagonals in any measurement count as 1 square. Beyond that each diagonal counts as 2 squares. The result is that at typical engagement ranges in the rules (3-7 squares) the warping of a measurement is limited compared to a true straight line measurement. I know it sounds odd but it really works (I made some drawings in a graphics program to test this out when I read about it in the PBI rules). You get more warping at the extremes but big deal. To me it's the relative values that matter most.
This is related to your second question about ranges. Yes, if each square in a "5:1" game is 200m or 300m then counting from one square to the next can in theory represent a broader array of ranges (start digression...my rules are really 3:1 or 4:1...5:1 makes no sense to me since the vast majority of weapons and vehicles are organized in multiples of roughly 3:1 or so...the 5:1 organization of German and US tank platoons in WWII was the exception rather than the rule and designers such as Frank Chadwick have said that it was these two company organizations that guided them to the 5:1 ratio...end digression) But that's not unlike the problem of using a ruler and having a target completely out of range because it's 1/4 inch beyond the shooter's max range band. In fact, the square approach is probably more representative of reality in this case since in the real ** CENSORED ** world ranges are more fungible than in the strict ruler-model typically seen in a wargame (ie shooting at a target 3 squares away better represents the fact that in the real world rounds don't suddenly fall out of the sky simply because a target is beyond some absolute range value).
As to pixelated terrain, you are very much correct from a technical perspective. The status of each square is binary in that it's either one type of square or another. But as to the look of the game (something that's important to me) the presentation need not be blocky. In my experiments I actually found the use of squares to be liberating in that very natural terrain can be made within a square but one need not worry about the implications on the game mechanics. This is especially true for urban terrain.
This brings up another benefit that I've discovered recently. Because terrain square definitions are binary all sorts of LOS and LOF issue become much easier to resolve at a glance and without reference to angle templates, rulers, stings-o-death, laser pointers, and the like. In the real world a soldier knows if he can see past the edge of a terrain feature. In traditional miniature games it becomes a class in geometry as gamers whip out their implements (which brings up yet another benefit...less table clutter with no rulers or templates!...the squares can be defined with very subtle yet distinct dots or even natural clumps or terrain such as weeds or gravel...unobtrusive yet very functional!).
This also goes to the positioning of stands within squares. With squares one can position the stand to look nice relative to the terrain in the square and not effect the outcome of the game. With the traditional ruler and template approach this is not possible. We've all been there before as a fellow gamer tries to precariously place a stand on some inconvenient terrain feature only to declare in frustration, "Well, you know where the stand really is" and the stand proceeds to slide down some miniature precipice. And this is fine and is what I've done but it's nice to place stands so they look good rather than tilted at some odd angle on a model terrain feature. My two little boys really clued me in to this issue when we played together as they placed their AFVs in urban squares to make them look like they were lurking through the city streets. This was much faster and far more colorful than the usual approach of placing the tiny vehicles "just so" and in adherence to a game's movement and LOS restrictions relative to the jumble of urban terrain features.
Anyway, the point of this thread was not really to debate the value of squares but to see if others used them in anyway. To me the use of squares (or hexes) now seems like a natural for microarmor. I was just curious as to how others might have used this approach in the past (eg Hauptmann's comments on hexes were very interesting.) That being said I can't say that the very reasonable objections that you and others have expressed are surprising since I had shared many of them myself until I tried it out! Everything written by every contributor so far has some degree of accuracy if not always based on actual experience which brings up the most salient point.
After playing a few test games I must admit that I too had hastily dismissed the idea but have really enjoyed how this approach lets me focus on the narrative of the game and its tactics rather than on the details of ruler measurements and template angles. But like I said this is all an experiment since I'm rather new to and still somewhat ignorant on the subject, so I'm VERY interested to read about other prespectives and experiences!
Tim
But today I did apply a grid to my best ground matt so I'm committed at this point at least for the cost of redoing my matt if this doesn't work!