= Solution
Use shifted degrees
$$
L_Y=\bigoplus_{j\ge1}\pi_{j+1}(Y)\otimes\mathbb Q
=\pi_*(\Omega Y)\otimes\mathbb Q.
$$
Its <graded Lie bracket> is the sign-adjusted <Whitehead bracket> from Question 2, equivalently the <Samelson product> on loops. This regrading is essential: the geometric bracket has degree minus one in the unshifted sphere grading. A <free graded Lie algebra> on a graded <vector space> $V$ means $\mathbb L(V)$ with the universal extension property for linear maps from $V$ to any <graded Lie algebra>.
Here are the general results used. The <Bott–Samelson theorem> says that, for a connected based <CW complex> $X$, the inclusion $X\to\Omega\Sigma X$ induces an isomorphism of graded algebras
$$
T(\widetilde H_*(X;\mathbb Q))\xrightarrow{\sim}
H_*(\Omega\Sigma X;\mathbb Q)
$$
with the <Pontryagin product>. The <Milnor–Moore theorem> says that for <simply connected> $Y$, rational loop <Hurewicz homomorphism> identifies $L_Y$ with the primitives of that loop-homology <Hopf algebra> and induces $U(L_Y)\cong H_*(\Omega Y;\mathbb Q)$. The graded <Poincaré-Birkhoff-Witt theorem> identifies the associated graded algebra of $U(L)$, for its word-length filtration, with the <graded symmetric algebra> on $L$; hence its <primitive elements of a Hopf algebra> are exactly $L$ over $\mathbb Q$. Finally the <rational Whitehead theorem> identifies <rational homology> equivalences of <simply connected> <CW complexes> with <rational homotopy equivalences>. <Rationalization of a topological space> induces the corresponding isomorphisms of loop-space <homotopy> and <rational homology>.
Suppose first $Y$ has the <rational homotopy type> of $\Sigma X$. A <reduced suspension> is <simply connected> when $X$ is connected; a disconnected $X$ has $H_1(\Sigma X;\mathbb Q)=\widetilde H_0(X;\mathbb Q)\ne0$, so it cannot have the rational homology of the simply connected $Y$. Put $V=\widetilde H_*(X;\mathbb Q)$, in strictly positive degrees. <Bott–Samelson theorem> gives $H_*(\Omega Y;\mathbb Q)\cong T(V)$ as an algebra. To deduce freeness of the primitive <Lie algebra>, its coproduct must also be considered: the initial <Bott–Samelson theorem> generators can have reduced-diagonal terms when $X$ has nontrivial <cup products>.
By <Milnor–Moore theorem> write this <Hopf algebra> as $U(L_Y)$. Its algebra indecomposables satisfy
$$
Q(U(L_Y))\cong L_Y/[L_Y,L_Y],\qquad Q(T(V))\cong V.
$$
The first identity follows from the enveloping relation: after all positive products are killed, every bracket is killed and no additional linear relation is introduced. Hence each homogeneous <basis> vector of $V$ has a primitive representative in $H_*(\Omega Y;\mathbb Q)$. Choose such representatives $p_v$. Sending $v$ to $p_v$ defines an algebra map from $T(V)$ to itself whose effect on indecomposables is the identity. It is an isomorphism: $p_v=v+$ decomposable terms, and every factor in those terms has smaller positive degree than $v$. Degree induction solves recursively for each original generator and constructs the inverse. This works even with arbitrarily many generators in a degree.
Equip the source <tensor algebra> with primitive generators. The map is now a Hopf-algebra isomorphism, because the chosen images are primitive. The primitives in this tensor <Hopf algebra> are exactly $\mathbb L(V)$: $U(\mathbb L(V))=T(V)$ by the two universal properties, and the graded <Graded Poincaré–Birkhoff–Witt theorem> gives its primitive subspace. Consequently $L_Y\cong\mathbb L(V)$, proving the forward direction without assuming a primitive coproduct for the original generators.
Conversely suppose $L_Y\cong\mathbb L(W)$ for a positively graded $W$. Choose a homogeneous <basis> $w_\lambda$ of $W$. Represent its corresponding <homotopy class> by a based sphere map $S^{|w_\lambda|+1}\to Y_{\mathbb Q}$, and combine these maps into
$$
F:Z=\bigvee_\lambda S^{|w_\lambda|+1}\longrightarrow Y_{\mathbb Q}.
$$
The <wedge sum> is a <reduced suspension>, namely $Z=\Sigma(\bigvee_\lambda S^{|w_\lambda|})$. The preceding direction, applied to this <wedge sum>, gives its free rational <homotopy> <Lie algebra> on the sphere inclusions. Indeed these particular tensor generators are primitive because the reduced diagonal of each sphere is zero, and loop <Hurewicz homomorphism> sends the adjoint sphere inclusion to that generator. Naturality of the bracket shows that $F_*$ sends the free generating <basis> to the prescribed free generating <basis> of $L_Y$. It is therefore an isomorphism of <free graded Lie algebras>, hence an isomorphism in every rational <homotopy> degree. The <rational Whitehead theorem> makes $F$ a <rational homotopy equivalence>. This proves the reverse implication and also shows that a <wedge sum> of spheres can be chosen as the <reduced suspension>.
For every chosen connected $X$ with $Y\simeq_{\mathbb Q}\Sigma X$, the required algebra statement is precisely the <Bott–Samelson theorem> isomorphism, transported along <rational homotopy equivalence>:
$$
\boxed{H_*(\Omega Y;\mathbb Q)\cong
T(\widetilde H_*(X;\mathbb Q))
=\bigoplus_{r\ge0}\widetilde H_*(X;\mathbb Q)^{\otimes r}.}
$$
Multiplication is concatenation of tensors and the degree-zero tensor is the unit. All freeness assertions here are graded, with the Koszul signs described above.
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