diff --git a/pandas/_libs/reduction.pyx b/pandas/_libs/reduction.pyx
index fa9c12777eb5b..ab99771385962 100644
--- a/pandas/_libs/reduction.pyx
+++ b/pandas/_libs/reduction.pyx
@@ -493,8 +493,8 @@ def apply_frame_axis0(object frame, object f, object names,
         object piece
         dict item_cache
 
-    if frame.index._has_complex_internals:
-        raise InvalidApply('Cannot modify frame index internals')
+    # We have already checked that we don't have a MultiIndex before calling
+    assert frame.index.nlevels == 1
 
     results = []
 
@@ -625,7 +625,7 @@ def compute_reduction(arr, f, axis=0, dummy=None, labels=None):
 
     if labels is not None:
         # Caller is responsible for ensuring we don't have MultiIndex
-        assert not labels._has_complex_internals
+        assert labels.nlevels == 1
 
         # pass as an ndarray/ExtensionArray
         labels = labels._values
diff --git a/pandas/core/groupby/ops.py b/pandas/core/groupby/ops.py
index ae397277de41c..aa925c5aea022 100644
--- a/pandas/core/groupby/ops.py
+++ b/pandas/core/groupby/ops.py
@@ -159,27 +159,26 @@ def apply(self, f, data, axis: int = 0):
             com.get_callable_name(f) not in base.plotting_methods
             and hasattr(splitter, "fast_apply")
             and axis == 0
-            # with MultiIndex, apply_frame_axis0 would raise InvalidApply
-            # TODO: can we make this check prettier?
-            and not sdata.index._has_complex_internals
+            # apply_frame_axis0 doesn't allow MultiIndex
+            and not isinstance(sdata.index, MultiIndex)
         ):
             try:
                 result_values, mutated = splitter.fast_apply(f, group_keys)
 
-                # If the fast apply path could be used we can return here.
-                # Otherwise we need to fall back to the slow implementation.
-                if len(result_values) == len(group_keys):
-                    return group_keys, result_values, mutated
-
             except libreduction.InvalidApply as err:
-                # Cannot fast apply on MultiIndex (_has_complex_internals).
-                # This Exception is also raised if `f` triggers an exception
+                # This Exception is raised if `f` triggers an exception
                 # but it is preferable to raise the exception in Python.
                 if "Let this error raise above us" not in str(err):
                     # TODO: can we infer anything about whether this is
                     #  worth-retrying in pure-python?
                     raise
 
+            else:
+                # If the fast apply path could be used we can return here.
+                # Otherwise we need to fall back to the slow implementation.
+                if len(result_values) == len(group_keys):
+                    return group_keys, result_values, mutated
+
         for key, (i, group) in zip(group_keys, splitter):
             object.__setattr__(group, "name", key)
 
@@ -615,7 +614,7 @@ def agg_series(self, obj: Series, func):
             # TODO: is the datetime64tz case supposed to go through here?
             return self._aggregate_series_pure_python(obj, func)
 
-        elif obj.index._has_complex_internals:
+        elif isinstance(obj.index, MultiIndex):
             # MultiIndex; Pre-empt TypeError in _aggregate_series_fast
             return self._aggregate_series_pure_python(obj, func)