Invariably generators would be great, but I mean just generators. Do you know some argument to explain it (in the former case)?

@YCor oh, I see.

@YCor What is "right" in this context?

@YCor The $1$-convergent condition is also essential to prove the expected theorems I think. But I had not thought about the possible non existence before. It is a good point!

@BenjaminSteinberg that makes sense. I wrote a sketch of this idea to convince myself.

@BenjaminSteinberg I agree (this need to agree with the definition of free profinite group as a particular case). My question is because Ribes, in his book, defines it using sheaves (which Neukirch call bundles I think). Perhaps this is the same thing after all, but in principle it seems quite different and unnatural to me.

@YCor the original statement says that there exists $i$ and $g \in G$ ($G$ is the free product) such that $F^g \subset G_i$.

@YCor the paper is: link.springer.com/article/10.1007/BF02567982

@HJRW of course. It was a careless mistake.