Here are a few additional thoughts, based on your “I’ve tried many, many matchups, but they all have thus far failed” comment and on aardvarkpepper’s analysis, which among other things raises the question of whether the kind of matrix you’re contemplating actually achieves the fairness it’s supposed to deliver.
To illustrate the argument I’m about to make, I’ll use the relatively simple case of the three-player matrix, which has already been solved, my version of it being:
A B C
1 2 3
4 1 5
6 7 1
2 6 4
7 5 2
3 4 7
5 3 6
The first point to consider is this. You’ll note that every player does indeed play in each position (A, B and C) exactly once, and that every player plays against every other player exactly once, as required. At first glance, everything looks perfectly symmetrical, and therefore perfectly “fair”, because the assumption being made is that “symmetry equals fairness.” It turns out, however, that there are two problems here. Problem one has already been pointed out by Aardvarkpepper: the concept that “symmetry equals fairness” is questionable. Problem two is that the matrix is only symmetrical when you view it it terms of every player playing against every other INDIVIDUAL player. It stops being symmetrical when you view it it terms of every player playing against every possible COMBINATION of players. In the above example, for instance, 1 gets to play against three combinations of players; 2&3, 4&5, and 6&7, but never gets to play against all the other possible combinations of players (such as 2&6). If you work from the assumption that “symmetry equals fairness”, the logical conclusion would be that your players would all have to play a game against all the other possible combinations of players, not just against every individual player. This would not only increase the number of games to be played, it also require you to drop the requirement that each play play each position only once.
The second point to consider is this. As noted, the above matrix does not include all the possible three-player combinations. Some of the possible combinations are “valid”, in the sense that they work within the matrix, while others are “invalid”, in the sense that they wreck the matrix. This may explain the “I’ve tried many, many matchups, but they all have thus far failed” problem you’ve been running into. Your five-player matrix experiments presumably all use the following match-ups as their starting points, since those match-ups are quick and easy to identify…
1 2 3 4 5
1 6 7 8 9
1 10 11 12 13
1 14 15 16 17
1 18 19 20 21
2 6 10 14 18
2 7 11 15 19
2 8 12 16 20
2 9 13 17 21
…but I’m wondering if those match-ups include “invalid” ones that make the rest of the matrix impossible to complete? I’m not saying that your contemplated 5-player matrix is impossible; I’m saying that working it out may require a good deal of mathematical knowledge (which I certainly don’t have). If you look at the Wikipedia articles on round-robin tournaments…
https://en.wikipedia.org/wiki/Round-robin_tournament
https://en.wikipedia.org/wiki/Tournament_(graph_theory)
…you’ll note that the scheduling algorithms for even the relatively simple case of rotating two-player match-ups are quite complicated, or at least (to borrow a phrase from Calvin and Hobbes) look complicated “to the untutored eye of the ignorant layman.” This doesn’t appear to be a problem you’ll be able to solve by continuing to “diligently working on at least finding ways that DON’T work for the 5-player, 21-participant bracket.”
Which brings me to a practical suggestion. The problem you’re working on is extremely difficult to solve if the only solution that’s acceptable to you is one that meets your requirements perfectly. If you settle for “almost perfectly,” however, it immediately becomes quite manageable. I assume from your remark “I’ve tried many, many matchups, but they all have thus far failed” that every time you try to set up a matrix, there are always one or two match-ups that don’t fit. How about simply living with them? As has ben discussed above, the premise that “symmetry equals fairness” has got a couple of conceptual problems with it, so it seems to me that having a couple of non-fitting outliers in your matrix is hardly going to be a fatal flaw. And you’ll note that the Wikipedia articles mention the concept “dummy competitors”, whose function seems to be make scheduling algorithms work. If the professionals need to use loopholes of this type, there’s no dishonour in your doing likewise.