For a further introduction to extremal graph theory, see Chapters 7 and 10 of the book Graph Theory by Reinhard Diestel.
The Hajnal-Szemerédi theorem can actually be phrased in the language of equitable colorings rather than clique factors. A short proof of the theorem was given by Kierstead and Kostochka;
The rotation-extension method (used to prove Dirac's theorem) can give very elegant proofs for Hamilton cycle problems. However, it often is not a viable proof option, particularly in the setting of directed graphs.
In this case the regularity lemma has been widely applied. An example is available in Hamiltonian degree sequences in digraphs.
The Lo-Markström lemma presented is a special case of Lemma 1.1 from F-factors in hypergraphs via absorption (which is stated more generally about hypergraphs).
Some further applications of this lemma can be found in this paper also.
For some more background on the regularity lemma see Chapters 7 and 9 of the book Graph Theory by Reinhard Diestel. In particular, the regularity lemma is also useful in situations
other than embedding (almost) spanning subgraphs. An appication in Ramsey theory can be found in Chapter 9.
Further background on epsilon-regular pairs and the regularity lemma can be found in an old thesis of mine.