Stable extender contract via `fit` / `_fit` resp `predict` / `_predict` separation

### Describe the workflow you want to enable

tl;dr, I am suggestion to refactor `scikit-learn` internals to a layer separation with boilerplate between `fit` and `_fit` resp `predict` and `_predict` methods, to make extender interfaces more stable. Also see https://github.com/scikit-learn/scikit-learn/issues/31728

More background: Currently, every time `scikit-learn` releases a new minor version - e.g., 1.5.0, 1.6.0, 1.7.0 - compliant extensions, e.g., custom transformers, classifiers, etc, break - specifically, referring to the API conformance as tested through `check_estimator` or `parametrize_with_checks`.

These repeated breakages in the "extender contract" contrast the stability of the usage contract, which is stable and professionally managed.

For a package like `scikit-learn` which means to be a standard not just for ML algorithms but also an API standard that everyone uses, this is not a good state to be in - "do not break user code" is the maxim that gets broken for power users writing extensions.

Of course maintaining downwards compatibility is not always possible, but nothing should break without a proper warning.

### Describe your proposed solution

The `fit`/`_fit` separation would ensure stability of the extension contract - and would also allow to build secondary deprecation patterns in relation to it.

The (oop) pattern this would implement is the so-called "template pattern".

It would allow to remove likely changing parts such as the boilerplate (e.g., `validate_data` vs `_validate_data` and such) from the extension locus, and thus completely prevent breakage in relation to boilerplate changes.
Reference: https://refactoring.guru/design-patterns/template-method

Examples of how this can be used to improve stability:

* `sktime`, for a different API, has a separation between `fit` calling an internal `_fit`, where change-prone boilerplate is sandwiched between a stable user contract (`fit`) and a stable extender contract (`_fit`); similarly `predict` and `_predict`
* `feature-engine` overrides the `BaseTransformer` `scikit-learn` extension contract with a similar pattern using `super()` calls in `fit` etc.

The proposed `fit`/`_fit` pairing that combines strategy and template pattern can be introduced easily via pure internal refactoring - this also add a layer of tests that checks the stability of the extension contract.

As a side effect, this would likely reduce the amount of copy-pasted boilerplate among the core estimators in `scikit-learn` proper as well.

### Describe alternatives you've considered, if relevant

The obvious alternative is "do nothing" - which will lead to packages with extensions producing their own version of mitigating patterns for the unstable status quo.

The problem with this is that the extending packages are mutually incompatible with their extender pattern. It would be better if there is a single stable extension pattern prescribed - or at least recommended - by `scikit-learn`.

### Additional context

_No response_

COMMENT:
> Note that we should be careful considering changing private methods as breaking change. It'd be impossible to develop the library if we can't change our private tooling. However, even in those cases, we've sometimes introduced a (short) deprecation cycle.

Agreed. Some might say that this is the purpose of private methods :D I thought the problem we were trying to solve is that too many things are marked as private (and as a result can change without prior warning). The "too many" is in the sense that many well behaved third-parties end up relying on private things. The goal of this issue is to move some of that to a more well defined "developer API", I think.

Labels: RFC, Developer API