This PR revives #26744.

This creates a new system: Convergence. See #24128.

I'm putting this up now so that I can document some TODOs. Because it's in a messy state, I plan to squash everything into one commit before merging.

Tasks that remain:

• Consider any necessary changes for the checkConvergence() API. Currently I have iteration index as the one and only argument, since we may want to use the same convergence class for multiple types of convergence check, and different types of convergence check will need to obtain that index differently. The same argument can be applied to others. Also I think that some classes will only be compatible with certain types of convergence check. We would probably want to do some checking to make sure it's used in a compatible way. Perhaps we should have some kind of IterationType argument? I'm open to other ideas.
  UPDATE: Opened Make Convergence object checks aware of solve type #28927 for this task.
• Understand the role of EquationSystems - does it do anything to set convergence parameters in there? Should Convergence being doing that? Some parameters in ResidualConvergence are unclear on if they're actually used, like the linear solve parameters. They get set in the EquationSystems object (which might be eventually set to PETSc), but at least some do not seem to come back into the class in the convergence check, maybe only getting used in libMesh's own convergence checks?
  UPDATE 1: The EquationSystems object is populated with the convergence-related parameters and then used in various convergence checks of different solvers, which do not all have the freedom to specify a generic convergence criteria. My plan is to not interact with the EquationSystems object from the Convergence classes if it is not necessary, and let the Executioner parameters populate it, as they do currently.
  UPDATE 2: The eigen executioners modify convergence-related parameters through EquationSystems, so it looks like we'll need to work with EquationSystems in the Convergence object after all.
• Throw an error if the user supplies any shared convergence parameter to the executioner when they supply a ResidualConvergence? The current behavior is to allow the executioner parameters but override them. That's maybe the simplest implementation-wise, but seems very error-prone for users.
  UPDATE: I plan to give only a warning if the user specifies nonlinear convergence parameters in the executioner when they have a non-default convergence.
• Improve PostprocessorConvergence to perform checks of execute_on. Here the Convergence would need to know what kind of iterations it will be used with, so we need to think about how to do this.
  Update: Opened Perform checks on execute_on for PostprocessorConvergence #28928 for this task.
• Address any relevant comments from Convergence system #26744.

I opened another issue related to this: #28936.