I've had the same problem everywhere I've worked and no company has even tried.  If it's just you making the accounts, just keep a spreadsheet or separate database on an...

Could possibly be Kerberos SPNs needing to be created for the new server.

The SQL Server service account often needs SPNs created to make a conneciton, or you get anonymous NTLM...

USE <dbname>

GO

GRANT EXECUTE TO <username>

This grants execute to all stored procedures and functions at database level.  No database role is created.