You can put a club on the ground behind the ball. Heck, you can use your putter to mark your ball's position if you want.
I'm not sure why this is confusing.
You can't point to a spot on the green and touch it to indicate the line of the putt. You can't have an alignment aid present when making a stroke. You can't test the surface of the green.
Other than that, what other rules could possibly prevent you from setting your putter down on the green? Heck, you can set another club down behind your ball if you want, so long as you move it before you make your stroke.
BTW, rather than do whatever you're doing, get some ball markers like this and you won't need to use your putter:
It's confusing when it comes to trying to find the words in the rules that conclusively prove your statements. I believe your assessment is correct, but you're relying on ideas not defined in the rules (at least the excerpts we've seen in this thread or that I can recall off the top of my head).
The rule that could prevent you is the bit about touching the green to indicate the line for putting. I believe (though don't know for certain) that this would include using your putter or another club to extend your reach. What in the rules tells us that it's ok to touch the green by lying the putter on the ground behind the ball for alignment, which certainly could be considered indicating the line for putting.
Again, I don't think your statements are wrong, and I think the interpretation would allow this. The confusing part is figuring out whether this is actually stated in the rules, or if it's simply leaning on common sense/common practice.