Non-living entity CustomName is shown to players, potentially containing debug or sensitive information

The bug

In 23w41a a change was made that caused the CustomName of non-living entities to display to players when looked at. This is problematic because this custom name might've been used to store debug information or even sensitive information, with no way to disable showing this custom name, even on entities existing before this version.

How to reproduce

1. Summon a non-living entity (like a minecart) with a custom name

   /summon minecart ~ ~ ~ {CustomName:'"debug_info_custom_minecart"'}

2. ❌ Notice that the custom name is visible, even to survival or non-opped players

Expected behavior

The custom name on non-living entities (which was private before) should not be visible, if necessary with an opt-out but preferably with an op-in to show the name.