No. Both technically and politically.
The technical problem is that the "NBN guy" visited your premises to install a new cable between the node and your house and to install a new socket inside your house. This is usually done by tying the new cable onto the old cable in the street, then pulling the old cable out from the wall socket until the new cable appears inside the house. With the old cable cut at an unknown location it cannot be used to pull the new cable.
The political problem is that you do not have a first socket. It sounds as though it was removed illegally, or you would have been in the Telstra database as having no existing connection. Now that the installer has completed a job report on the unsuccessful installation, you are in the NBN database as having no existing connection, and are very likely to be classified as requiring a new installation not a transfer from POTS to NBN. Which is a whole different (and likely much slower) process.
Your question actually appears to be asking whether you can run VDSL over Cat3. Technically the answer is yes if you want a kilobit per second data connection. Politically the answer is no, as you will then have an illegal connection.