diff --git a/Lib/fontTools/subset/__init__.py b/Lib/fontTools/subset/__init__.py
index 4b9cb00f6..7f08709ef 100644
--- a/Lib/fontTools/subset/__init__.py
+++ b/Lib/fontTools/subset/__init__.py
@@ -15,7 +15,6 @@ from fontTools.subset.util import _add_method, _uniq_sort
 from fontTools.subset.cff import *
 from fontTools.subset.svg import *
 from fontTools.varLib import varStore  # for subset_varidxes
-from fontTools.ttLib.tables._n_a_m_e import NameRecordVisitor
 import sys
 import struct
 import array
@@ -2586,10 +2585,25 @@ def prune_post_subset(self, font, options):
 
     if self.version == 1:
         kept_labels = []
+        dropped_labels = []
         for i, label in enumerate(self.paletteEntryLabels):
             if i in retained_palette_indices:
                 kept_labels.append(label)
+            else:
+                dropped_labels.append(label)
         self.paletteEntryLabels = kept_labels
+        # Remove dropped labels from the name table.
+        name_table = font["name"]
+        name_table.names = [
+            n
+            for n in name_table.names
+            if (
+                n.nameID not in dropped_labels
+                # Only remove nameIDs in the user range and if they're not explicitly kept
+                or n.nameID < 256
+                or n.nameID in options.name_IDs
+            )
+        ]
     return bool(self.numPaletteEntries)
 
 
@@ -2901,10 +2915,32 @@ def prune_pre_subset(self, font, options):
 
 
 @_add_method(ttLib.getTableClass("name"))
-def prune_post_subset(self, font, options):
-    visitor = NameRecordVisitor()
-    visitor.visit(font)
-    nameIDs = set(options.name_IDs) | visitor.seen
+def prune_pre_subset(self, font, options):
+    nameIDs = set(options.name_IDs)
+    fvar = font.get("fvar")
+    if fvar:
+        nameIDs.update([axis.axisNameID for axis in fvar.axes])
+        nameIDs.update([inst.subfamilyNameID for inst in fvar.instances])
+        nameIDs.update(
+            [
+                inst.postscriptNameID
+                for inst in fvar.instances
+                if inst.postscriptNameID != 0xFFFF
+            ]
+        )
+    stat = font.get("STAT")
+    if stat:
+        if stat.table.AxisValueArray:
+            nameIDs.update(
+                [val_rec.ValueNameID for val_rec in stat.table.AxisValueArray.AxisValue]
+            )
+        nameIDs.update(
+            [axis_rec.AxisNameID for axis_rec in stat.table.DesignAxisRecord.Axis]
+        )
+    cpal = font.get("CPAL")
+    if cpal and cpal.version == 1:
+        nameIDs.update(cpal.paletteLabels)
+        nameIDs.update(cpal.paletteEntryLabels)
     if "*" not in options.name_IDs:
         self.names = [n for n in self.names if n.nameID in nameIDs]
     if not options.name_legacy:
@@ -3427,14 +3463,7 @@ class Subsetter(object):
             font.setGlyphOrder(self.new_glyph_order)
 
     def _prune_post_subset(self, font):
-        tableTags = font.keys()
-        # Prune the name table last because when we're pruning the name table,
-        # we visit each table in the font to see what name table records are
-        # still in use.
-        if "name" in tableTags:
-            tableTags.remove("name")
-            tableTags.append("name")
-        for tag in tableTags:
+        for tag in font.keys():
             if tag == "GlyphOrder":
                 continue
             if tag == "OS/2":
diff --git a/Lib/fontTools/ttLib/tables/_n_a_m_e.py b/Lib/fontTools/ttLib/tables/_n_a_m_e.py
index 152e4f268..517f7d39f 100644
--- a/Lib/fontTools/ttLib/tables/_n_a_m_e.py
+++ b/Lib/fontTools/ttLib/tables/_n_a_m_e.py
@@ -242,9 +242,6 @@ class table__n_a_m_e(DefaultTable.DefaultTable):
             if record.nameID not in visitor.seen:
                 toDelete.add(record.nameID)
 
-        if not toDelete:
-            return
-        log.info(f"Deleting name records with NameIDs {toDelete}")
         for nameID in toDelete:
             ttFont["name"].removeNames(nameID)
         return toDelete