I agree that a Turing Machine can do "all the possible mathematical problems". But that is because it is just a machine representation of an algorithm: first do this, then do that, finally output that.

I mean anything that is solvable can be represented by an algorithm (because that is precisely the definition of 'solvable'). It is just a tautology. I said nothing new here.

And by creating a machine representation of an algorithm, that it will also solve all the possible problems is also nothing new. This is also mere tautology. So basically when it is said that Turing Machine is the most powerful machine, what it effectively means is that the most powerful machine is the most powerful machine!

Definition of "most powerful": That which can accept any language. Definition of "Algorithm" : Process of doing anything Machine representation of "Algorithm" : A machine that can do anything 

Therefore it is only logical that the machine representation of algorithm will be the most powerful machine. What's the new thing Alan Turing gave us?