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I also believe that even if one is interested in smooth manifolds, it is much easier and conceptual (or perhaps natural) to define "smooth" manifolds through diffeological spaces, just like the definition of topological manifolds. Actually, we can say that a smooth manifold is a diffeological space that is locally Euclidean: every point in the space has an open neighborhood (with respect to the D-topology) which is diffeomorphic to an open subset of a fixed Euclidean space. After that, one can add Hausdorfness and second-countability requirements, if needed. In this situation, there is no need to talk about the smooth compatibility of charts, because it is automatically verified.

I also believe that even if one is interested in smooth manifolds, it is much easier and conceptual (or perhaps natural) to define "smooth" manifolds through diffeological spaces, just like the definition of topological manifolds. Actually, we can say that a smooth manifold is a diffeological space that is locally Euclidean: every point in the space has an open neighborhood (with respect to the D-topology) which is diffeomorphic to an open subset of a fixed Euclidean space. After that, one can add Hausdorfness and second-countability requirements, if needed. In this situation, there is no need to talk about the smooth compatibility of charts, because it is automatically verified. Notice that diffeological spaces are an extension of Euclidean spaces in the first place.

I also believe that even if one is interested in smooth manifolds, it is much easier and conceptual (or perhaps natural) to define "smooth" manifolds through diffeological spaces, just like the definition of topological manifolds. Actually, we can say that a smooth manifold is a diffeological space that is locally Euclidean: every point in the space has an open neighborhood (with respect to the D-topology) which is diffeomorphic to an open subset of a fixed Euclidean space. After that, one can add Hausdorfness and second-countability requirements, if needed. In this situation, there is no need to talk about the smooth compatibility of charts, because it is automatically verified.

• 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

• The (internal) tangent space of the diffeomorphism group of a compact manifold 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