Get rid of `OrdinaryDiffEq` internals usage

[SouthEndMusic]

#1761 introduced the function residual which makes use of OrdinaryDiffEq internals. This should be removed when possible, probably when the upstream issue is fixed: SciML/OrdinaryDiffEq.jl#2442.

[SouthEndMusic]

The PR attempt for this was closed. We eventually want to use the NonlinearSolveAlg API, but that is not mature enough yet, see SciML/OrdinaryDiffEq.jl#2671 (comment).