Your comments are a bit confusing to follow.. but Apple's iOS does not take up an entire phone worth of storage. I am running the latest iOS and my 'System' storage is only 6.71 GB. Even the old 16 GB phones can handle that size. Now there have been some issues with the System and Other categories becoming inflated for unknown reasons. When you have that issue, it can fill up your storage pretty quickly. I'm pretty sure your phone will not download something if it does not have space.
The resolution for the issue above at this point is to backup your device if possible, reset and restore it. So as Lawrence Finch has commented, you still need to have a current/recent backup when you do this. Best to do this regularly and frequently as you never know when you'll need it.
If you are being prompted to buy more than 200 GB of cloud storage, it must be because you have gotten close to filling it up. What are you storing in iCloud? Assuming your new phone is signed into the same Apple ID, if you go to Settings and tap your Apple ID up top, then select iCloud.. it will show you what apps are using iCloud, what space you're taking up and what's still available.
I just want to make sure that you aren't under the impression that just because you have a 256 GB phone, that you automatically need as much iCloud storage.