Operating open trays with photochemistry in complete darkness is not something I would recommend to anyone except under the most special circumstances. If safelights are a concern, I'd recommend leaving a test strip out there for 5-10 minutes, then develop&fix. If this test strip turns out noticeably darker than a test strip going directly into developer&fixer, then yes, safelights do need closer attention.
Rudi,
Using photo chemistry in open trays in total darkness is standard operating procedure for developing sheet film in trays. I do it all the time

Doing it once in a while as a safelight check or the like is easy-peasy.
... I must admit I don't see any point whatsoever to the test described in the original post, in relation to print quality. It only tests for one specific thing, and there is no defined purpose. The test does not establish "development to completion", nor does it consider tonality at the various development times. ... I'm still at a loss with respect to the purpose of this test. What is it supposed to tell you? What is it for? ...
Such a test is to determine "how long is too long," i.e., when the paper starts to exhibit fog from the developer. In my experience, this point is fairly long for FB papers; somewhere in the 6-10 minute range. As for "developing to completion," Gerald explains it above admirably. I think of it also as four phases:
1. induction (when the paper sits there and then finally a faint image starts to emerge),
2. contrast building or "reaching final curve shape" (the real developing happens here, and for most papers, the curve shape is reached at the end of this stage and doesn't change after that; max black is attained and the final contrast is established.)
3. A period in which the extended development causes the curve to move to the right on the chart, in essence, the same effect as giving a bit more print exposure. I use this to fine-tune exposure all the time. Fifteen extra seconds of development gives a noticeable difference at development times between two and four minutes. This is helpful when I've already got tricky dodging and burning down and just need a bit more general exposure; instead of exposing more, I'll just develop a bit more. But...
4. After a certain time in the developer, the whites begin to fog. At this point, contrast decreases by being compressed from the fog; highlight separation and max white suffer.
This latter is undesirable and what the OP is testing for. It seems to me, however, that not only would it be different from paper to paper and batch to batch, but would also vary with the age of the paper. It might be a good test to do to check, however, if you are using relatively long development times and suspect fog for a particular paper at that time.
That raises this question: what is the point of development to completion? I know that I should not leave paper in the developer for too long, so what is too long and how do I determine what is too long?
Doc, See the last paragraph above. However, mostly what we want to do is avoid underdevelopment. Only when the paper has reached the point where the curve shape has been established and maximum black and contrast are possible is the print really fully developed. The manufacturer gives you this time, however, so just use it if in doubt. Extending development will give an effect just like adding more exposure till the fog point is reached. This seems to be what you're testing for.
Stop the presses!
Hexavelent, a local Apugger, came by to help and was suspicious of the results so far. He said the test strips looked like they were printing out. He suspected that the fixer was bad - the SECOND batch of fixer - so we put a piece of film in it just to check. Nothing. The film didn't budge. That fixer might as well have been gin (which I surely need about now). ...
It's always good to do tests with fresh chemistry

I'm curious to know what the problem was with your fixer, since you indicate that you mixed a fresh batch of fix for the second round of tests. Any ideas? Stock stored too long?
BTW, your "fog" was just the uncleared emulsion that was slowly turning brown due to exposure to light. Now you know what happens to prints when the fixer is spent
Best,
Doremus