In the particular case of surfaces, I found the following reference which includes a proof that is not too complicated: Regular Mappings and the Space of Homeomorphisms on 2-Manifolds by Hamstrom and Dyer. They prove local contractibility, which is more than I asked. It works for surfaces with or without boundaries and includes a slight generalization with fixed points on the boundary. This is Theorem 1 in this article. The proof fits in 6 pages, is a bit heavy on notations but this remains managable. Unfortunately there is no figure. The proof uses conformal maps for a couple of lemmas, Alexander's trick, and a technique due to J.H. Roberts but I cannot find the corresponding reference. They also cite a German article of Kneser as a source of inspiration, but my knowledge of German is very basic so reading it would represent quite an investment.

In the particular case of surfaces, I found the following reference which includes a proof that is not too complicated: Regular Mappings and the Space of Homeomorphisms on 2-Manifolds by Hamstrom and Dyer. They prove local contractibility, which is more than I asked. It works for surfaces with or without boundaries and includes a slight generalization with fixed points on the boundary. This is Theorem 1 in this article. The proof fits in 6 pages, is a bit heavy on notations but this remains managable. Unfortunately there is no figure. The proof uses conformal maps for a couple of lemmas, Alexander's trick, and a technique due to J.H. Roberts (Local arcwise connectivity in the space $H^n$ of homeomorphisms of $S^n$ onto itself, Summary of Lectures, Summer Institute on Set Theoretic Topology, Madison, Wisconsin, 1955, p. 100) but I cannot find the corresponding reference. They also cite a German article of Kneser (Die Deformationssätze der einfach zusammenhägenden Flächen, Mathematische Zeitschrift, Vol. 25(1926), pp. 362-372) as a source of inspiration, but my knowledge of German is very basic so reading it would represent quite an investment.

In the particular case of surfaces, I found the following reference which includes a proof that is not too complicated: Regular Mappings and the Space of Homeomorphisms on 2-Manifolds by Hamstrom and Dyer. It works for surfaces with or without boundaries and includes a slight generalization with fixed points on the boundary. This is Theorem 1 in this article. The proof fits in 6 pages, is a bit heavy on notations but this remains managable. Unfortunately there is no figure. The proof uses conformal maps for a couple of lemmas, Alexander's trick, and a technique due to J.H. Roberts but I cannot find the corresponding reference. They also cite a German article of Kneser as a source of inspiration, but my knowledge of German is very basic so reading it would represent quite an investment.

In the particular case of surfaces, I found the following reference which includes a proof that is not too complicated: Regular Mappings and the Space of Homeomorphisms on 2-Manifolds by Hamstrom and Dyer. They prove local contractibility, which is more than I asked. It works for surfaces with or without boundaries and includes a slight generalization with fixed points on the boundary. This is Theorem 1 in this article. The proof fits in 6 pages, is a bit heavy on notations but this remains managable. Unfortunately there is no figure. The proof uses conformal maps for a couple of lemmas, Alexander's trick, and a technique due to J.H. Roberts but I cannot find the corresponding reference. They also cite a German article of Kneser as a source of inspiration, but my knowledge of German is very basic so reading it would represent quite an investment.