diff --git a/distributed/tests/test_worker.py b/distributed/tests/test_worker.py
index 29b789618c7..6c1d73f00bd 100644
--- a/distributed/tests/test_worker.py
+++ b/distributed/tests/test_worker.py
@@ -1595,7 +1595,6 @@ async def f(ev):
     assert s.tasks["f1"].state in ("queued", "no-worker")
 
 
-@pytest.mark.slow
 @gen_cluster(client=True, nthreads=[("", 1)])
 async def test_close_async_task_handles_cancellation(c, s, a):
     ev = Event()
@@ -1622,6 +1621,8 @@ async def f(ev):
     task.cancel()
     await asyncio.wait({task})
 
+    await a.close(timeout=1)
+
 
 @pytest.mark.slow
 @gen_cluster(client=True, nthreads=[("", 1)], timeout=10)
diff --git a/distributed/worker.py b/distributed/worker.py
index 27dbc5166f2..7a4e2adbc3d 100644
--- a/distributed/worker.py
+++ b/distributed/worker.py
@@ -29,7 +29,7 @@
 from concurrent.futures import Executor
 from contextlib import suppress
 from datetime import timedelta
-from functools import wraps
+from functools import wraps, partial
 from inspect import isawaitable
 from typing import (
     TYPE_CHECKING,
@@ -2304,17 +2304,11 @@ async def execute(self, key: str, *, stimulus_id: str) -> StateMachineEvent:
             try:
                 ts.start_time = time()
                 if iscoroutinefunction(function):
-                    token = _worker_cvar.set(self)
-                    try:
-                        result = await apply_function_async(
-                            function,
-                            args2,
-                            kwargs2,
-                            self.scheduler_delay,
-                        )
-                    finally:
-                        _worker_cvar.reset(token)
-                elif "ThreadPoolExecutor" in str(type(e)):
+                    coro = function(*args2, **kwargs2)
+                    function = partial(asyncio.run, coro)
+                    args2 = ()
+                    kwargs2 = {}
+                if "ThreadPoolExecutor" in str(type(e)):
                     # The 'executor' time metric should be almost zero most of the time,
                     # e.g. thread synchronization overhead only, since thread-noncpu and
                     # thread-cpu inside the thread detract from it. However, it may
@@ -3157,54 +3151,6 @@ def apply_function_simple(
     return msg
 
 
-async def apply_function_async(
-    function,
-    args,
-    kwargs,
-    time_delay,
-):
-    """Run a function, collect information
-
-    Returns
-    -------
-    msg: dictionary with status, result/error, timings, etc..
-    """
-    ident = threading.get_ident()
-    try:
-        with context_meter.meter("thread-noncpu", func=time) as m:
-            result = await function(*args, **kwargs)
-    except (SystemExit, KeyboardInterrupt):
-        # Special-case these, just like asyncio does all over the place. They will pass
-        # through `fail_hard` and `_handle_stimulus_from_task`, and eventually be caught
-        # by special-case logic in asyncio:
-        # https://github.com/python/cpython/blob/v3.9.4/Lib/asyncio/events.py#L81-L82
-        # Any other `BaseException` types would ultimately be ignored by asyncio if
-        # raised here, after messing up the worker state machine along their way.
-        raise
-    except BaseException as e:
-        # NOTE: this includes `CancelledError`! Since it's a user task, that's _not_ a
-        # reason to shut down the worker.
-        # Users _shouldn't_ use `BaseException`s, but if they do, we can assume they
-        # aren't a reason to shut down the whole system (since we allow the
-        # system-shutting-down `SystemExit` and `KeyboardInterrupt` to pass through)
-        msg = error_message(e)
-        msg["op"] = "task-erred"
-        msg["actual-exception"] = e
-    else:
-        msg = {
-            "op": "task-finished",
-            "status": "OK",
-            "result": result,
-            "nbytes": sizeof(result),
-            "type": type(result) if result is not None else None,
-        }
-
-    msg["start"] = m.start + time_delay
-    msg["stop"] = m.stop + time_delay
-    msg["thread"] = ident
-    return msg
-
-
 def apply_function_actor(
     function, args, kwargs, execution_state, key, active_threads, active_threads_lock
 ):