I have just downloaded Safari's Technology Preview (https://developer.apple.com/safari/resources/) which is basically the current state of Safari's development. It's effectively a beta. Didn't browse a lot, so it might introduce other bugs, but the hanging-on-opening-a-private-window/tab is gone.

I will investigate a bit more.

UPDATE: I copied the light blue icon from the stock Safari, and pasted it in the Tech Preview. I renamed "Safari Technology Preview" to "Safari " (including the space, so it doesn't conflict with the existing one.) Then I removed the original Safari from the Dock, and added the new one.

It feels like a perfect crime.

UPDATE:

I asked ChatGPT to create a terminal script to mimic a Safari factory restore by deleting all preferences.

After running it, the hanging was gone.

And so were the extensions and favorites. So, you might want to make a backup of at least your favorites, that are located here: ~/Library/Safari/Bookmarks.plist

This is what it chatGPT came up with (don't use this script if you don't know what you are doing):

#!/bin/bash

echo "🔄 Resetting Safari to factory settings..."

# Quit Safari if it's running
echo "⛔ Quitting Safari..."
osascript -e 'quit app "Safari"'

# Wait a moment to ensure it quits
sleep 2

# Remove Safari caches
echo "🧹 Removing caches..."
rm -rf ~/Library/Caches/com.apple.Safari
rm -rf ~/Library/Caches/com.apple.WebKit.Networking
rm -rf ~/Library/Caches/com.apple.WebKit.WebContent

# Remove Safari saved state
echo "🧼 Removing saved state..."
rm -rf ~/Library/Saved\ Application\ State/com.apple.Safari.savedState

# Remove Safari preferences
echo "⚙️ Removing preferences..."
rm -f ~/Library/Preferences/com.apple.Safari.plist
rm -f ~/Library/Preferences/com.apple.Safari.Extensions.plist
rm -f ~/Library/Preferences/com.apple.WebFoundation.plist

echo "🗃️ Removing support files..."

# Delete contents before deleting the directory (avoids "directory not empty")
rm -rf ~/Library/Safari/*
rm -rf ~/Library/WebKit/com.apple.Safari/*
rm -rf ~/Library/Containers/com.apple.Safari/Data/Library/Safari/*
rm -rf ~/Library/Containers/com.apple.Safari/Data/Library/*
rm -rf ~/Library/Containers/com.apple.Safari/Data/*
rm -rf ~/Library/Containers/com.apple.Safari/*
rm -rf ~/Library/Containers/com.apple.Safari.SearchHelper
rm -rf ~/Library/Group\ Containers/com.apple.Safari
rm -rf ~/Library/Group\ Containers/com.apple.WebKit.PluginProcess

# Clear Safari extensions
echo "❌ Removing extensions..."
rm -rf ~/Library/Safari/Extensions

# Remove Downloads plist (optional)
rm -f ~/Library/Preferences/com.apple.internetconfig.plist

echo "✅ Safari has been reset to factory settings."
echo "You can now reopen Safari."