I have over a 100 gb of storage free on iCloud, yet my phone still says it doesn’t have storage to work properly

The message refers to the physical storage space on the phone itself. Having more iCloud storage won’t help with this. You need to delete content on the phone in order to free up space.

See this for more information.

What's the difference between device storage and iCloud storage? - Apple Support

iCloud storage is separate from your phone storage. Phone storage capacity is FIXED and cannot be added to. If our phone wants more storage you will need to delete items from it or move items to offline storage