Bump compat for DiffEqBase.jl and DiffEqCallbacks.jl

[JoshuaLampert]

Let's already have a look in this PR if CI passes, when we remove the compat bounds.

In conjunction with #2150:
Closes #1789, closes #1785, closes #1240, closes #2108, closes #2120, closes #2121, closes #2157, closes #1791, closes #1850, closes #2091.

[github-actions]

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[codecov]

Codecov Report

All modified and coverable lines are covered by tests ✅

Project coverage is 96.44%. Comparing base (9f2a5be) to head (5248ed1).
Report is 2 commits behind head on main.

Additional details and impacted files

@@            Coverage Diff             @@
##             main    #2263      +/-   ##
==========================================
- Coverage   96.87%   96.44%   -0.43%     
==========================================
  Files         490      490              
  Lines       39462    39462              
==========================================
- Hits        38228    38057     -171     
- Misses       1234     1405     +171     

Flag	Coverage Δ
unittests	96.44% <ø> (-0.43%)	⬇️

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[JoshuaLampert]

The good news is that the new versions of the SciML stack are used (the latest versions of DiffEqBase.jl, SciMLBase.jl, OrdinaryDiffEq.jl, DiffEqCallbacks.jl) and everything precompiles successfully. The bad news is that we still have some work to do to fix some errors mostly related to SciML packages.

[JoshuaLampert]

Closing this in favor of #2264.