How do I transfer data from a broken phone without using cloud storage?

With a broken phone, the only chance you have of restoring your data is from a computer backup or iCloud backup. If you did have a computer backup of your previous phone, you can restore that data to your new phone by following the steps here and if there was an iCloud backup you will also find those steps.

Restore your iPhone, iPad, or iPod touch from a backup - Apple Support

If you prefer to use a computer backup and not store data in iCloud, then this will help you to prevent data from being stored in iCloud, and turn it off altogether.

Change which apps sync and store data with iCloud - Apple Support

"How do I transfer data from a broken phone without using cloud storage? [...]For some reason nothing synced over even with the phones plugged in together and plugged into computer. I personally do not like cloud storage. What are the other options to get the data from my old phone.[...]"

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Transferring to New iPhone without QuickStart:

When transferring to a new iPhone, QuickStart is what to use. But, being damaged, you can't have that choice. So, The transfer may have stalled, or you may have run out of iCloud Storage (in which case, you'd upgrade your iCloud Storage Plan). So, if you enabled items in iCloud on the old phone, then they would have synced to iCloud. So, you'll need to enable those same items in iCloud. To do so...

1. Open: Settings
2. Tap: Name
3. Tap: iCloud
4. Tap: See All button
5. Tap: [an item]
6. Toggle On: Use this on iPhone