classical mechanics - Why the involution condition is imposed in the definition of integrability?

For an $N$-degree-of-freedom system to be integrable, the usual definition imposes the existence of $N$ independent conserved quantities, which must be in involution to each other, i.e.,

$$\{ F_i, F_j \} ~=~ 0\tag{1}$$

Why this extra condition? Why is

$$\{ H, F_i \}~=~0 \tag{2} $$

insufficient?

Answer

I) Well, the existence of the involution

$$\{ I_i, I_j \}_{PB} ~=~ 0,\qquad i,j\in\{1, \ldots, n\}.\tag{1}$$

is already generically true locally for a Hamiltonian system, cf. e.g. this Phys.SE post. So it is essentially only a global/topological requirement in the definition of (Liouville) integrability.

II) Another possible answer (in light of OP's second equation) is that we would like to write the Hamiltonian $H=H(I_1, \ldots, I_n)$ in terms of the action variables $I_1, \ldots, I_n$ alone [i.e. without the angle variables, which in turn should be cyclic/ignorable variables]. Then the involution (1) implies OP's second condition

$$ \dot{I}_i~=~\{ I_i,H \}_{PB} ~=~ 0,\qquad i\in\{1, \ldots, n\}, \tag{2}$$

i.e. that the action variables $I_1, \ldots, I_n$ are constants of motion [if the system is autonomous and the action variable contain no explicit (but possibly implicit) time dependence]. In other words, a failure of (1) could jeopardize (2).

III) Finally it seems relevant to mention that the involution (1) is the main assumption that goes into the Caratheodory–Jacobi–Lie theorem, which guarantee that the $n$ actions variables $I_1, \ldots, I_n$ locally can be extended into a full set of $2n$ Darboux coordinates, aka. canonical coordinates. See also e.g. this related Phys.SE post.