Yes, the structure you describe forms a model structure. However, some care is needed. First, topological groups are not algebras over an operad, because operads don't encode inverses. Topological monoids are algebras over the operad $Ass$, and you can use Denis Nardin's idea in that case to transfer the model structure from simplicial monoids, or you can use David Roberts's idea to transfer from topological spaces (by which I mean compactly generated spaces, because otherwise you won't get a closed symmetric monoidal category of spaces). This story goes back to Schwede and Shipley's paper Algebras and Modules in monoidal model categories, and Equivalences of monoidal model categories.
Topological groups are algebras over an algebraic, or Lawvere, theory. The homotopy theory of such algebras goes back to Rezk, Bergner, and Badzioch. The last paper starts off with topological groups as the motivating example. All three of those papers are written simplicially (meaning, technically, they produce model structures on simplicial groups), but I just had a look at Rezk's paper and it seems his proof goes through mutatis mutandis for topological spaces. It's Theorem 7.2, and the only hard part is Lemma 7.6, which relies on a path space lifting argument that was also used by Berger and Moerdijk in Axiomatic homotopy theory for operads and given perhaps its most general form in Section 5 of this paper of mine with Donald Yau. The last point is that the domains of the generating trivial cofibrations in Top (hence in TopGr) are small with respect to inclusions of topological spaces (see chapter 2 of Hovey's book), and transfinite compositions and pushouts preserve inclusions.
