Can I attribute the conservation of mechanical energy to the power of maths, especially vector calculus?
Is this the true story behind the process during which people discover conservation of mechanical energy?
No, energy conservation was historically a long debate over centuries.
Is there a better way of arriving at the conclusion?
For that you should start with a Lagrangian, possibly with some time symmetry, then learn how to get dynamics from the Lagrangian. Then find out there is a constant associated with that time symmetry, so that even though dynamics means things change there is something that doesn't, something associated with that symmetry in time.
Then we can go a step further. And ask our self the reason the Lagrangian depended on time when it depended on time. And the answer truly is because something else changed, and if something is changing then it has its own dynamics. So if Lagrangian mechanics is the way the world is, then there should be a Lagrangian for the larger system.
And there is. And that Lagrangian determines the dynamics for that larger system, and then with the time symmetry, there is a conserved quantity.
So when there was a symmetry it indicated we had a closed system, evolving on its own. When the Lagrangian didn't have that symmetry it indicated an external interaction and energy was not conserved because of the interaction.
This idea of open and closed system of external and internal energy and interactions is totally key. And the idea that you have failed to truly describe the total system if energy is not conserved.
That is the true story of energy conservation. It is aspirational. It becomes like Newton's third law a judgement on a theory that if found wanting makes the theory seem incomplete.
Sure you could have an external force but it will seem incomplete. And you can increase or decrease your total energy but that will also seem incomplete.
That said: energy is not actually conserved in the world we live in.
When you include fields all you really get is a kind of local conservation law for energy density. And local conservation doesn't mean global conservation if your space is flat and bounded or if it not flat. And to get energy from energy density you need volume, hence geometry. And once you bring geometry into it things get an additional complication.
First, energy is geometrically just part of energy-momentum and breaking it into four pieces is as arbitrary as picking a direction for a coordinate axis. And further, enrrgy-momentum itself has a flux as well as a density. So you have an entire stress-energy tensor for it all.
And at this point potential energy has finally died a quiet death because now the energy flows and the energy sits throughout space instead of being stored abstractly in the idea of things being part of a system or not a system.
So no, potential energy is not the proper way to understand conservation of energy. It doesn't correctly assign energy a place or a correct flow. It can make it seem like energy is conserved when in our universe energy is not conserved.