chore(tracing): refactor web server integrations to use the core module (#12035)

Refactors all web server integrations still using `tracer.trace` to
instead use `core.context_with_data`. This is in preparation for
supporting AWS API Gateway to ensure all web servers share the same code
path.

- [x] PR author has checked that all the criteria below are met
- The PR description includes an overview of the change
- The PR description articulates the motivation for the change
- The change includes tests OR the PR description describes a testing
strategy
- The PR description notes risks associated with the change, if any
- Newly-added code is easy to change
- The change follows the [library release note
guidelines](https://ddtrace.readthedocs.io/en/stable/releasenotes.html)
- The change includes or references documentation updates if necessary
- Backport labels are set (if
[applicable](https://ddtrace.readthedocs.io/en/latest/contributing.html#backporting))

- [x] Reviewer has checked that all the criteria below are met
- Title is accurate
- All changes are related to the pull request's stated goal
- Avoids breaking
[API](https://ddtrace.readthedocs.io/en/stable/versioning.html#interfaces)
changes
- Testing strategy adequately addresses listed risks
- Newly-added code is easy to change
- Release note makes sense to a user of the library
- If necessary, author has acknowledged and discussed the performance
implications of this PR as reported in the benchmarks PR comment
- Backport labels are set in a manner that is consistent with the
[release branch maintenance
policy](https://ddtrace.readthedocs.io/en/latest/contributing.html#backporting)

Author: wconti27

ddtrace/contrib/internal/aiohttp/middlewares.py

@@ -2,6 +2,7 @@
 from aiohttp.web_urldispatcher import SystemRoute
 
 from ddtrace import config
+from ddtrace.contrib.asyncio import context_provider
 from ddtrace.ext import SpanTypes
 from ddtrace.ext import http
 from ddtrace.internal import core
@@ -59,9 +60,8 @@ async def attach_context(request):
             request[REQUEST_CONFIG_KEY] = app[CONFIG_KEY]
             try:
                 response = await handler(request)
-                if not config.aiohttp["disable_stream_timing_for_mem_leak"]:
-                    if isinstance(response, web.StreamResponse):
-                        request.task.add_done_callback(lambda _: finish_request_span(request, response))
+                if isinstance(response, web.StreamResponse):
+                    request.task.add_done_callback(lambda _: finish_request_span(request, response))
                 return response
             except Exception:
                 req_span.set_traceback()