Jupyter-packaging was outdated; nowadays, Jupyter uses Hatch.

https://github.com/jupyter/jupyter-packaging

UPD:
I understand why Jupyter is starting to use it, mainly because it supports shared data, which makes it super easy to share JavaScript, HTML, etc., files.

There is also another section, [tool.jupyter-releaser.hooks], which seems important if you want to perform a release. The question is, why do we also need to release to Jupyter, instead of just making it possible to install from PyPI? What are the benefits of adding it there as well?"

https://jupyter-releaser.readthedocs.io/en/latest/get_started/making_release_from_repo.html#prep-release

When an extension is being used, the method _get_labextension_metadata is called. This method requires the _jupyter_labextension_paths method. Without it, the extension will not be imported successfully.

Therefore, you need to include some code like this in your __init__.py file:

import json
from pathlib import Path

__version__ = "0.1.0"

HERE = Path(__file__).parent.resolve()

with (HERE / "labextension" / "package.json").open() as fid:
    data = json.load(fid)

def _jupyter_labextension_paths():
    return [{
        "src": "labextension",
        "dest": data["name"]
    }]

Here is the _get_labextension_metadata code for reference:

for package in package_candidates:
    try:
        m = importlib.import_module(package)
        if hasattr(m, "_jupyter_labextension_paths"):
            return m, m._jupyter_labextension_paths()
    except Exception as exc:
        errors.append(exc)

msg = f"There is no labextension in {module}. Errors encountered: {errors}"
raise ModuleNotFoundError(msg)