I'm pretty sure there doesn't exist a "slicker" proof of this fact in the literature. The proof you describe exists in many forms starting with Dehn and Lickorish -- as I said in a comment, the particular arrangement of it you gave (making use of the complex of curves) is basically due to Ivanov. The only fundamentally different approach I know of is in the paper
Generating the mapping class group (an algebraic approach). (English summary)
Publ. Mat. 40 (1996), no. 2, 457–468.
This proof is purely algebraic (and rather more complicated than the topological proof). It is based on McCool's 1975 proof that the mapping class group is finitely presentable. By the way, McCool is often written out of the history of the subject, but his work predates Hatcher-Thurston and is the first paper in the literature that explicitly proves that the mapping class group is finitely presentable (though the algebraic geometers proved equivalent facts in the early '60's).
I tend to view the fact that the mapping class is generated by Dehn twists as equivalent to the fact that the complex of curves is connected. For this, there exist several alternate proofs. Many of these proofs also give higher connectivity; I'll try to indicate that as I go.
There is the combinatorial approach taken in many sources (eg Farb-Margalit's "Primer on Mapping Class Groups"). I don't know who to attribute this to. You could probably prove that the curve complex is simply-connected by this method, but I doubt you could prove higher connectivity.
You could use Teichmuller theory -- this is how Harer originally proved that the curve complex is highly connected.
There is a Morse-theoretic proof due to Ivanov (it is described in his survey "Mapping Class Groups"). It is an adaption of the Cerf theory approach to proving the mapping class group is finitely presentable which is due to Hatcher and Thurston. In his paper "Complexes of curves and Teichmüller modular groups", Ivanov uses this approach to prove that the complex of curves is highly connected.
You could easily deduce that the curve complex is connected from Hatcher's slick proof that the arc complex is contractible, which is located in his paper "On triangulations of surfaces". I don't think they have this written up, but Hatcher and Vogtmann have a proof that the curve complex is highly connected with this as the starting point.