It turns out(Edit: my original remark didn't take all of the question into account. One can say that the Kronecker factor doesn'tdoes retain a sufficient amount of information about the correlations $\mu(A\cap T^{-n}A\cap T^{-2n}A)$detail to prove the desired result. Indeed, the Kronecker factor is characteristic for the averages, but $\lim_{N\to\infty} \frac{1}{N}\sum_{n=1}^N \mu(A\cap T^{-n}A\cap T^{-2n}A)$ can be smaller than $\mu(A)^3$ (unfortunately I don't have an example in mind at the moment).
The large intersection property you ask for was first established by Bergelson, Host, and Kra [1]. Frantzikinakis [2] gave a simplified proof and extended the result to polynomial configurations. Ackelsberg, Bergelson, and Best [3] have extended the result to actions of countable abelian groups, and the introduction there has a nice overview of the history and technical issues involved. I believe [2] and [3] use the method you describe.
[1] Bergelson, Vitaly; Host, Bernard; Kra, Bryna, Multiple recurrence and nilsequences (with an appendix by Imre Ruzsa), Invent. Math. 160, No. 2, 261-303 (2005). ZBL1087.28007.
[2] Frantzikinakis, Nikos, Multiple ergodic averages for three polynomials and applications, Trans. Am. Math. Soc. 360, No. 10, 5435-5475 (2008). ZBL1158.37006.
[3] Ackelsberg, Ethan; Bergelson, Vitaly; Best, Andrew, Multiple recurrence and large intersections for abelian group actions, ZBL07471818.