baggage and context are ignored. setting values is ignored.

Describe your environment

$ pip list | grep opentelemetry
opentelemetry-api                        1.23.0
opentelemetry-exporter-otlp-proto-common 1.23.0
opentelemetry-exporter-otlp-proto-http   1.23.0
opentelemetry-proto                      1.23.0
opentelemetry-sdk                        1.23.0
opentelemetry-semantic-conventions       0.44b0
$ python -V
Python 3.10.5

Steps to reproduce

from opentelemetry import baggage, context, trace
from opentelemetry.sdk.resources import Resource
from opentelemetry.sdk.trace import TracerProvider
from opentelemetry.sdk.trace.export import ConsoleSpanExporter
from opentelemetry.sdk.trace.export import SimpleSpanProcessor


class TestTelemetry:
    def test_it(self):
        # Create provider and get tracer
        resource = Resource.create({'service.name': 'test-python-1'})
        provider = TracerProvider(resource=resource)

        exporter = ConsoleSpanExporter()
        processor = SimpleSpanProcessor(exporter)
        provider.add_span_processor(processor)

        trace.set_tracer_provider(provider)
        tracer = trace.get_tracer('shell-energy-kafka-python')

        # Set context values
        exec_ctx = context.get_current()
        print('exec_ctx:', exec_ctx)

        print('context.set_value("greetings", "hello world")')
        key = 'greetings'
        value = 'hello world'
        context.set_value(key, value)
        print('exec_ctx:', context.get_current())

        value_from_get = context.get_value(key)
        print(f'exec_ctx: key:"{key}", value:"{value_from_get}"')

        # Start span
        attrs = {'extra.attr': 'extra.attr.value'}
        with tracer.start_as_current_span('Testing traces',
                                          kind=trace.SpanKind.PRODUCER,
                                          attributes=attrs):
            # Set baggage
            baggage_key = 'greetings_bag'
            baggage_value = 'hello world bag'

            print(f'set_baggage: key:"{baggage_key}", value="{baggage_value}"')
            baggage.set_baggage(baggage_key, baggage_value)

            print('baggage.get_all():', baggage.get_all())

Output:

exec_ctx: {}
context.set_value("greetings", "hello world")
exec_ctx: {}
exec_ctx: key:"greetings", value:"None"
set_baggage: key:"greetings_bag", value="hello world bag"
baggage: {}
{
    "name": "Testing traces",
    "context": {
        "trace_id": "0x7c50f96a522f16b6633a8ca827a12551",
        "span_id": "0x4eeef4315fee0241",
        "trace_state": "[]"
    },
    "kind": "SpanKind.PRODUCER",
    "parent_id": null,
    "start_time": "2024-03-15T12:04:13.600564Z",
    "end_time": "2024-03-15T12:04:13.600588Z",
    "status": {
        "status_code": "UNSET"
    },
    "attributes": {
        "extra.attr": "extra.attr.value"
    },
    "events": [],
    "links": [],
    "resource": {
        "attributes": {
            "telemetry.sdk.language": "python",
            "telemetry.sdk.name": "opentelemetry",
            "telemetry.sdk.version": "1.23.0",
            "service.name": "test-python-1"
        },
        "schema_url": ""
    }
}

What is the expected behavior?
I expect baggage.get_all() to return keys and values previously set in baggage. I expect the same from the context.
I also expect the values set in baggage and in context will appear in the trace printed to console.

What is the actual behavior?

1. baggage.get_all() and context.get_current() return empty objects.
2. Trace printed in the console contains none of the values set on baggage and on context.

Additional context
None

Reproduce the behavior with the provided Python snippet, then read the context.set_value, context.get_current, baggage.set_baggage, and baggage.get_all entry points. Check how values are expected to be propagated into start_as_current_span and exported spans; done means the documented behavior is covered by tests or the issue's expectations are clarified.