Is it possible to extract data from a damaged iPhone 13 Pro without repairing it?

If the phone is so badly damaged that you cannot use it, and that it will not even synchronize with Finder/iTunes, there is no way to retrieve your data from the phone. The only way you are going to get your data back is if you had copies of it somewhere other than on that iPhone, before the phone was run over.

Were you using iCloud synchronization? Or making backups that might include some of your data (and could be restored to another phone)? Or copying your data off the phone in other ways? If not, it's gone.

apparently would cost around £500 to repair front and back?

Apple will have to inspect the device first, but I doubt that they will even offer repairs on the device.

Is there anyway to extract data

No, even if Apple does offer repairs on the phone

No

But I won’t probably be able to do that with the phone screen being disabled?

We'll try to make this as clear as possible.

You won't be able to do anything with the phone unless / until it is repaired.....if.....repairs are possible.

But even if you have the phone repaired, you will not be able to retrieve the current data on the phone.

So, no matter what you decide......the data on the phone cannot be retrieved.

We previously answered above......

Is there anyway to extract data

No, even if Apple does offer repairs on the phone

But, you must have missed that. Glad that you now have a clear understanding.