(Edit: my original remark didn't take all of the question into account. One can say that the Kronecker factor does retain sufficient detail to prove the desired result.)
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.