SQL Server DB Restore Unable to browse directory
First thing you need to check is that:

Has the SQL Server service account been added as a member of any local
groups on EACH server? Does the group in which it resides have read-write
permissions?

So please check if you're NW service account is admin on the box, along with R/W permissions for it to access the folder for backups.

Also, in case you are on cluster,

then you need to make sure and check for the dependency of that drive when in cluster for the service account its running under!