As far as I am aware, you can find some applications of diffeology "merely" in differential geometry of manifolds in the following list (I am not sure this list is exhaustive):
The (internal) tangent space of the diffeomorphism group of a compact manifold at the identity is the space of its vector fields, See [Hector G. Géométrie et topologie des espaces difféologiques. Analysis and geometry in foliated manifolds (Santiago de Compostela, 1994). 1995 Nov 17:55-80.] or [J.D. Christensen, E. Wu, Tangent spaces and tangent bundles for diffeological spaces, Cah. Topol. G'{e}om. Diff'{e}r. Cat'{e}g. 57(1) (2016), 3-50.].
The diffeomorphism group of a Lie foliation, See [Hector G, Macías-Virgós E, Sotelo-Armesto A. The diffeomorphism group of a Lie foliation. InAnnales de l'Institut Fourier 2011 (Vol. 61, No. 1, pp. 365-378).]
De Rham cohomology of diffeological spaces and foliations, See [Hector G, Macías-Virgós E, Sanmartín-Carbón E. De Rham cohomology of diffeological spaces and foliations. Indagationes Mathematicae. 2011 Aug 1;21(3-4):212-20.]
The basic de Rham complex of a singular foliation, See [Miyamoto D. The Basic de Rham Complex of a Singular Foliation. International Mathematics Research Notices.]
Basic forms and orbit spaces: a diffeological approach, See [Karshon Y, Watts J. Basic forms and orbit spaces: a diffeological approach. SIGMA. Symmetry, Integrability and Geometry: Methods and Applications. 2016 Mar 8;12:026.]
The orientation-preserving diffeomorphism group of $\mathbb{S}^2$ deforms to SO (3) smoothly, See [Li J, Watts JA. The orientation-preserving diffeomorphism group of $\mathbb{S}^2$ deforms to SO (3) smoothly. Transformation Groups. 2011 Jun;16(2):537-53.]
Smooth Lie group actions are parametrized diffeological subgroups, See [Iglesias-Zemmour P, Karshon Y. Smooth Lie group actions are parametrized diffeological subgroups. Proceedings of the American Mathematical Society. 2012 Feb;140(2):731-9.]
Differential forms on manifolds with boundary and corners See [Gürer S, Iglesias-Zemmour P. Differential forms on manifolds with boundary and corners. Indagationes Mathematicae. 2019 Sep 1;30(5):920-9.]
The Geodesics of the 2-Torus See here.
Every symplectic manifold is a (linear) coadjoint orbit See [Donato P, Iglesias-Zemmour P. Every symplectic manifold is a (linear) coadjoint orbit. Canadian Mathematical Bulletin. 2022 Jun;65(2):345-60.]
De Rham cohomology for the complement of a (dense) irrational flow on the torus See this MO post and Patrick I-Z's answer