Open questions/known issues with current implementation of direct-hybrid method for aeroacoustics

This whole process seems to be overly convoluted to me. It's a good prototype that works, so I wouldn't change it now, but it seems to me like we are abusing a lot of the OrdinaryDiffEq machinery here. We should somehow find another way to make this work in a cleaner fashion.

If not already done, can you maybe start an issue with open questions and known issues with this implementation, such that we can collect them in one place? First item from my side:
* Find a cleaner way to implement the coupled Euler-Acoustics evolution than manually using OrdinaryDiffEq internals

_Originally posted by @sloede in https://github.com/trixi-framework/Trixi.jl/pull/712#discussion_r669669399_

List of open questions and known issues:

* Find a cleaner way to implement the coupled Euler-Acoustics evolution than manually using OrdinaryDiffEq internals
* `.h5` files generated by `AveragingCallback` are currently not compatible with Trixi2Vtk (discussed [here](https://github.com/trixi-framework/Trixi.jl/pull/712#discussion_r702560088))

Start by reviewing the direct-hybrid implementation and its use of OrdinaryDiffEq internals, then read the linked pull-request discussions for context. Investigate the two listed concerns: a cleaner coupled Euler-Acoustics evolution and the incompatibility between AveragingCallback-generated .h5 files and Trixi2Vtk. Done means the open questions and known issues have been resolved with an agreed cleaner implementation and compatible output.

Written by the indexing model from the issue text.