ref(launchdarkly): Remove __init__ from integration

[sentrivana]

Description

The LaunchDarkly client is a singleton. As such, it doesn't make sense to provide a way to patch a specific client only.

This allows us to get rid of the integation __init__, which is always good: integrations are meant to be setup via setup_once.

Issues

Closes #7040

Reminders

• Please add tests to validate your changes, and lint your code using uv run ruff.
• Add GH Issue ID & Linear ID (if applicable)
• PR title should use conventional commit style (feat:, fix:, ref:, meta:)
• For external contributors: CONTRIBUTING.md, Sentry SDK development docs, Discord community

[github-actions]

Codecov Results 📊

✅ 90940 passed | ❌ 1 failed | ⏭️ 5436 skipped | Total: 96377 | Pass Rate: 94.36% | Execution Time: 334m 41s

📊 Comparison with Base Branch

Metric	Change
Total Tests	📉 -6836
Passed Tests	📉 -6034
Failed Tests	📈 +1
Skipped Tests	📉 -803

➕ New Tests (1)

View new tests

• test_trace_decorator_no_trx
  • File: tests.tracing.test_decorator
  • Status: ❌ Failing

❌ Failed Tests

test_trace_decorator_no_trx

File: tests.tracing.test_decorator
Suite: py3.10-gevent
Error: AssertionError: Expected 'mock' to be called once. Called 2 times. Calls: [call('Cannot create a child span for %s. Please start a Sentry transaction before calling this function.', 'test_decorator.my_example_function'), call('[Monitor] health check negative, downsampling with a factor of %d', 10)].

Stack Trace

tests/tracing/test_decorator.py:45: in test_trace_decorator_no_trx
    fake_debug.assert_called_once_with(
/usr/lib/python3.10/unittest/mock.py:940: in assert_called_once_with
    raise AssertionError(msg)
E   AssertionError: Expected 'mock' to be called once. Called 2 times.
E   Calls: [call('Cannot create a child span for %s. Please start a Sentry transaction before calling this function.', 'test_decorator.my_example_function'),
E    call('[Monitor] health check negative, downsampling with a factor of %d', 10)].

✅ Patch coverage is 100.00%. Project has 2334 uncovered lines.
✅ Project coverage is 90.15%. Comparing base (base) to head (head).

Coverage diff

@@            Coverage Diff             @@
##          main       #PR       +/-##
==========================================
+ Coverage    89.91%    90.15%    +0.24%
==========================================
  Files          193       186        -7
  Lines        24828     23692     -1136
  Branches      8930      8526      -404
==========================================
+ Hits         22322     21358      -964
- Misses        2506      2334      -172
- Partials      1416      1362       -54

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