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There is a very elegant and approach to derangements

D.M. Jackson, Laguerre polynomials and derangements, Math. Proc. Camb. Phil. Soc., 80 (1976), 213–214.

I am not sure it fits precisely your criteria, but I found Jackson's approach very useful for various asymptotic estimates.

show/hide this revision's text 1

There is a very elegant and approach to derangements

D.M. Jackson, Laguerre polynomials and derangements, Math. Proc. Camb. Phil. Soc., 80 (1976), 213–214.

I am not sure it fits precisely your criteria, but I found Jackson's approach very useful for various asymptotic estimates.