iPhone 14 ProMax won't update iOS 17.1, therefore cannot restore my new iPhone

You cannot pick and choose an iOS version. What you get is Settings > General > Software update > [All options are available here]

You need to correctly Identify your iPhone model - Apple Support

These are the iOS options based on an iPhone Model -->

• iOS 17.1.1 —>
  • A16 Bionic (4 nm) - 15 (6GB RAM) / 15 Plus (6GB RAM)
  • A17 Pro (3 nm) - 15 Pro (8GB RAM) / 15 Pro Max (8GB RAM)
  • Chipset A12 - XR (3GB RAM) / XS (4GB RAM)/ XS Max (4GB RAM)
  • Chipset A13 - SE (2020) (3GB RAM) / 11 (4GB RAM)/ 11 Pro (4GB RAM)/11 Pro Max (4GB RAM)
  • Chipset A14 - 12 Mini (4GB RAM) / 12 (4GB RAM) / 12 Pro (6GB RAM) / 12 Pro Max (6GB RAM)
  • Chipset A15 - SE (2022) (4GB RAM) / 13 Mini (4GB RAM) / 13 (4GB RAM) / 13 Pro (6GB RAM) / 13 Pro Max (6GB RAM)
  • Chipset A16 - 14 (6GB RAM)/ 14 Plus (6GB RAM) / 14 Pro (6GB RAM) / 14 Pro Max (6GB RAM)
• iOS 16.7.2 —>
  • Chipset A11 - 8 (2GB RAM) / 8 Plus (3GB RAM) / X (3GB RAM)
• iOS 15.8 —>
  • Chipset A9 - 6S (2GB RAM) / 6S Plus (2GB RAM) / SE (1st Gen) (2GB RAM)
  • Chipset A10 - 7 (2GB RAM) / 7 Plus (3GB RAM)
• iOS 12.5.7 —>
  • Chipset A7 - iPhone 5S (1GB RAM)
  • Chipset A8 - iPhone 6/6 Plus (1GB RAM)
• iOS 10.3.4 —>
  • Chipset A6 - iPhone 5 (1GB RAM)
• iOS 10.3.3 —>
  • Chipset A6 - iPhone 5C (1GB RAM)
• iOS 9.3.6 —> iPhone 4S (512MB RAM) - Chipset A5
• iOS 7.1.2 —> iPhone 4 (512MB RAM)- Chipset A4

Know how to...

• Update Over-the-Air (OTA) —> Update your iPhone or iPad - Apple Support (IN)
• Update using iTunes/Finder —> Update your iPhone, iPad, or iPod touch - Apple Support (IN)

lyellowbird1 Said:

"iPhone 14 ProMax must update to the latest version of iOS (iOS 17.1) before it can be restored, however when I try to connect to my PC (where my iPhone is backed up to), it comes up with message, 'There was problem downloading the software for the iPhone, The network connections timed out.' HELP!"

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Troubleshooting iPhone Updates:

Use a Computer and Ethernet:

Start with attempting an update through use of a computer. If you cannot use Wi-Fi, then use Ethernet.  See my User Tip: Update your iPhone Using a Computer - User Tip. On the computer, you would connect it through Ethernet, going directly into your router. For a Mac (or PC) that does not have Ethernet, this would be the adapter to use: Belkin USB-C to Gigabit Ethernet Adapter - Apple. If you get the same error, the model is likely just not compatible.