Best way to make wiring to outbuilding legal

I'm fixing some wiring in an older house and I believe the wiring to the outbuilding is not legal. I would like to correct it if possible, but will bring in a professional if necessary.

The current wiring is set up with 10-2 plus ground coming from a 30A breaker on the electrical box. It runs up the weather head where it is spliced to 6 gauge stranded overhead style wire with 2 blacks and a braided, unshielded ground. At the garage, it comes into the box and the unshielded ground is attached to the neutral bar, the hot is connected to the hot side of the box with a jumper to what should be the second leg of 240. The second black wire which is actually the neutral at the house is capped and the ground for the garage is an earth ground at the garage. Ground and neutral are not bonded at the box at least.

The handyman doing work on the house said to just throw a 240V 30A breaker into the house's box, disconnect the neutral (that is currently capped at the garage's box) and connect that wire to the new breaker then make the necessary fixes on the box in the garage. Assuming I mark the white wire to indicate it is hot, the garage would now have 2 legs of hot with neutral from the house along with earth ground at the garage. Would this be a legal fix?

Edit to add requested photos:

Wiring in outbuilding box showing the capped neutral

This shows how the 10-2 is attached to the outside wire