Replace old client with new storage api

[eqbech]

Issue
Resolves #14244

Approach
Short description of the approach

(Screenshot of new behavior in GUI if applicable)

• PR title captures the intent of the changes, and is fitting for release notes.
• Added appropriate release note label
• Commit history is consistent and clean, in line with the contribution guidelines.
• Make sure unit tests pass locally after every commit (git rebase -i main --exec 'just rapid-tests')

When applicable

• When screenshots are changed: Review screenshot-PR in ert-testdata,
  merge screenshot-PR in ert-testdata before merging this PR.
• When there are user facing changes: Updated documentation
• New behavior or changes to existing untested code: Ensured that unit tests are added (See Ground Rules).
• Large PR: Prepare changes in small commits for more convenient review
• Bug fix: Add regression test for the bug
• Bug fix: Add backport label to latest release (format: 'backport release-branch-name')

---

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[codecov-commenter]

❌ 1 Tests Failed:

Tests completed	Failed	Passed	Skipped
4259	1	4258	63

View the top 1 failed test(s) by shortest run time

tests/everest/test_everest_client.py::test_that_multiple_everest_clients_can_connect_to_server@math_func/config_minimal.yml

Stack Traces | 490s run time

cached_example = <function cached_example.<locals>.run_config at 0x7fdac91174c0>
change_to_tmpdir = None

    @pytest.mark.skip_mac_ci
    @pytest.mark.slow
    @pytest.mark.xdist_group("math_func/config_minimal.yml")
    @pytest.mark.flaky(rerun=3)
    @pytest.mark.skipif(
        sys.version_info[0:3] == (3, 13, 6), reason="Fails on Python 3.13.6"
    )
    def test_that_multiple_everest_clients_can_connect_to_server(
        cached_example, change_to_tmpdir
    ):
        # We use a cached run for the reference list of received events
        path, config_file, _, server_events_list = cached_example(
            "math_func/config_minimal.yml"
        )
    
        config_path = Path(path) / config_file
        config_content = yaml.safe_load(config_path.read_text(encoding="utf-8"))
        config_content["simulator"] = {"queue_system": {"name": "local", "max_running": 2}}
        config_path.write_text(
            yaml.dump(config_content, default_flow_style=False), encoding="utf-8"
        )
    
        ever_config = EverestConfig.load_file(config_path)
    
        # Run the case through everserver
        everest_main_thread = threading.Thread(
            target=everest_entry, args=[[str(config_path)]]
        )
    
        everest_main_thread.start()
>       api = ErtClient.for_project(
            Path(ServerConfig.get_session_dir(ever_config.output_dir))
        )

.../tests/everest/test_everest_client.py:194: 
_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 
.../ert/services/ert_client.py:102: in for_project
    return cls(SharedClient.get_client(project, connect_timeout), timeout)
               ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
.../ert/services/shared_client.py:58: in get_client
    client = create_ertserver_client(key, timeout=timeout)
             ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
.../ert/services/shared_client.py:140: in create_ertserver_client
    controller = create_ert_server_controller(timeout=timeout, project=project)
                 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 

project = PosixPath('.../everest-0/everest_output/.session')
timeout = 240, logging_config = None

    def create_ert_server_controller(
        *,
        project: os.PathLike[str],
        timeout: int | None = None,
        logging_config: str | None = None,
    ) -> ErtServerController:
        path = Path(project)
        # Wait for storage_server.json file to appear
        try:  # ruff: ignore[too-many-statements-in-try-clause]
            if timeout is None:
                timeout = 240
            t = -1
            while t < timeout:
                storage_server_path = path / _ERT_SERVER_CONNECTION_INFO_FILE
                try:
                    if storage_server_path.stat().st_size > 0:
                        storage_server_content = json.loads(
                            storage_server_path.read_text(encoding="utf-8")
                        )
    
                        return ErtServerController(
                            storage_path=str(path),
                            connection_info=storage_server_content,
                            logging_config=logging_config,
                        )
                except FileNotFoundError:
                    pass
    
                sleep(1)
                t += 1
    
>           raise TimeoutError("Server not started")
E           TimeoutError: Server not started

.../ert/services/ert_server.py:452: TimeoutError

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[ertomatic]

Screenshots differ from baselines. A baseline update PR has been prepared: equinor/ert-testdata#98

[ertomatic]

Screenshot tests now pass. The baseline update PR equinor/ert-testdata#98 has been closed.

[ertomatic]

Screenshot tests now pass. The baseline update PR equinor/ert-testdata#104 has been closed.

[ertomatic]

Screenshots differ from baselines. A baseline update PR has been prepared: equinor/ert-testdata#109

[ertomatic]

Screenshot tests now pass. The baseline update PR equinor/ert-testdata#109 has been closed.