Finite-Dynkin Hall algebras are generated by vertex simples

By [F1], every extension of $M_i^{\oplus a}$ by $M_j^{\oplus b}$ for $i\le j$ splits. Hence an ordered Hall product of such blocks is a nonzero scalar multiple of their direct sum: the only possible middle object is the direct sum, and at least its canonical summand inclusion contributes to the positive integer Hall count. Its Euler factor is a nonzero power of $\nu$. This proves that whenever the individual indecomposable classes in a sum have been generated, the class of that sum is generated. In particular $[S_i^{\oplus a}]$ is a nonzero scalar multiple of $[S_i{]}^a$, since there are no self-extensions at a loopless vertex.

Number the vertices topologically, with arrows from earlier to later vertices. For a dimension vector $d$, the product $[d_1{S}_1]\star \cdots \star [d_r{S}_r]$ has the same nonzero coefficient on every object of dimension $d$. Indeed it counts the unique descending filtration whose successive quotients are supported at these vertices: the subobject after each step is the entire representation on the remaining terminal vertex set and zero on the preceding vertices. This is a subrepresentation because arrows never go from a terminal vertex to an earlier one. There are no choices of subspaces within a vertex block. All Euler factors depend only on $d$, so their product is a common nonzero scalar. There are finitely many objects in the degree, since there are finitely many arrow matrices over $k$.

Induct on the total dimension of an indecomposable $M$. The one-dimensional case is a vertex simple. For larger dimension $d$, [F1] gives exactly one indecomposable of dimension $d$. Every other object of that degree is a direct sum of smaller-dimensional indecomposables by [F2]; its class is generated by induction and step 1.1. The simple word in step 2.1 is generated by step 1.1. Subtract all its other degree-$d$ classes and divide by its nonzero coefficient to isolate $[M]$. This proves generation of every indecomposable, and step 1.1 then proves generation of every object. The composition and extended conclusions are exactly their definitions in [F3]. Across distinct components the cross Euler form, Hom and Ext are zero, so the same argument covers every finite ADE forest, including the empty one.