chore: change asgi request span finish (#14774)

## Description

<!-- Provide an overview of the change and motivation for the change -->
Removes manually finishing `request_span` and adds to `trace_handlers`
to finish when context ends.

An extension of: [chore(asgi): refactor websocket code to use core
API](https://github.com/DataDog/dd-trace-py/pull/14345/files)

## Testing

<!-- Describe your testing strategy or note what tests are included -->

## Risks

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-->

## Additional Notes

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---------

Co-authored-by: Brett Langdon <[email protected]>

Author: quinna-h

ddtrace/_trace/trace_handlers.py

@@ -1262,6 +1262,7 @@ def listen():
         core.on(f"context.started.{context_name}", _start_span)
 
     for name in (
+        "asgi.request",
         "asgi.websocket.close.message",
         "asgi.websocket.disconnect.message",
         "asgi.websocket.receive.message",

ddtrace/contrib/internal/asgi/middleware.py

@@ -577,6 +577,3 @@ def _handle_websocket_disconnect_message(
             tags={COMPONENT: self.integration_config.integration_name, SPAN_KIND: SpanKind.CONSUMER},
         ) as ctx:
             core.dispatch("asgi.websocket.disconnect.message", (ctx, scope, message))
-
-        if request_span and request_span.error == 0:
-            request_span.finish()