Not having paper nor pencil at hand, the unconventional approach of using travelers came to me meanwhile seating at dentist office trying to entertain my mind in some absorbing subject and forget the upcoming procedures.
Now I realize my solution It basically is coincident with H. Eve , I enclose an sketch of it.
Certainly I need eleven travelers four for each vertex except for vertex A since I only have used five equations.

In my opinion for the original problem , there are two interesting facts as you point out no matter what size the first radius is, the midpoint of the two consecutive radii is always the tangency point of the incircle to the given triangle ABC and of course the length between the consecutive radii is invariant

The second is using the same sequence drawing circles in any polygon after the second round the value of the radii keep repeating and also the length between the consecutive radii is invariant.

Of course there are more interesting facts from the deep research you made like Gergonne- Adams Line and obviously all your second findings is a whole new prospective