Depending on your requirements, you might be able to keep things simpler and save money by moving away from Azure Files and instead using Blob containers.
https://learn.microsoft.com/en-us/azure/storage/blobs/storage-blobs-introduction
Current versions of SQL Server can backup directly to a blob URI.
Not sure what you are using to do the server backups and whether that software integrates with Azure Blob storage but if it doesn't, you could have the backup file created locally and script a job using AzCopy to move it up to Blob storage.
https://learn.microsoft.com/en-us/azure/storage/common/storage-use-azcopy-v10?tabs=dnf
There isn't anything built into Windows that I am aware of that is the equivalent of mapping blob storage as a convenient drive letter, but you could use Storage Explorer to interact with your backup data (supports drag and drop, copy and pasting of files and folders).
The advantage of Blob storage over Azure Files is cost, and the communications protocol is HTTPS 443 TLS 1.2 instead of SMB (TCP 445) which I am assuming won't be blocked by your ISP. If you were using fine grain permissions on files and folders in your SMB file share, you would have to adapt that to a container-based permissions model using Entra identities.
If you wanted to pursue the VPN path, it probably makes sense to create a private endpoint, for your Azure Files, in a subnet within the same VNET that the P2S VPN places you.
From your client computers, you would need to affect the DNS so that your xyz.file.core.... resolves to that private endpoint private IP address. If you don't create a private endpoint + DNS, when you hit xyz.file.core, it is still going to resolve to the public endpoint / IP address which will not traverse your tunnel. Your tunnel by default is probably only advertising and routing traffic destined to that VNET's IP space.
To quickly test this, you could create a private endpoint for the Azure File share and add an entry to your local host file and test it.