Can I use the following method to prove an algorithm is correct?
I am rather surprised that you raised this question since the meticulous and enlightening answers you have written to some math questions demonstrate sufficiently that you are capable of rigorous logical deduction. It seems that you became somewhat uncomfortable when you stumbled upon a new and unorthodox way to prove an algorithm is correct.

Believe in yourself. Believe in logic.  

Yes, I believe you have shown your algorithm is correct. To be absolutely clear, suppose you have shown all of the following.


An optimal solution exists. 
Your algorithm provides a solution and only one solution. 
Every solution that is not the solution provided by your algorithm cannot be optimal. 


Then your algorithm must provide an optimal solution. The implication is just simple logic.

You do not even need to show that solution provided cannot be improved in the same way as any other solution.