It is easy to find algebras T∈C in a finite tensor category C that naturally come with a lift to a braided commutative algebra T∈Z(C) in the Drinfeld center of C. In fact, any finite tensor category has at least two such algebras, namely the monoidal unit I and the canonical end ∫_X∈CX⊗X^∨. Using the theory of braided operads, we prove that for any such algebra T the homotopy invariants, i.e. the derived morphism space from I to T, naturally come with the structure of a differential graded E₂-algebra. This way, we obtain a rich source of differential graded E₂-algebras in the homological algebra of finite tensor categories. We use this result to prove that Deligne's E₂-structure on the Hochschild cochain complex of a finite tensor category is induced by the canonical end, its multiplication and its non-crossing half braiding. With this new and more explicit description of Deligne's E₂-structure, we can lift the Farinati-Solotar bracket on the Ext algebra of a finite tensor category to an E₂-structure at cochain level. Moreover, we prove that, for a unimodular pivotal finite tensor category, the inclusion of the Ext algebra into the Hochschild cochains is a monomorphism of framed E₂-algebras, thereby refining a result of Menichi.