Should I try to re-invent the wheel while doing research?
I don’t think this approach is feasible for the following reason:

Theoretical research (and any other research for that matter) takes much more time to perform than to understand.
For example, if you have sufficient mathematical background knowledge, I am pretty confident that a few hours suffice for me to make you fully appreciate and understand the entire (theoretical) research I did in the last year, such that the only difference between you and me would be that I have more experience with thinking in these my terms.
I am not very familiar with the history of quantum mechanics, but I am pretty confident that writing down the axioms was the result of quite some hard thinking, conceptualising, and a considerable amount of experience.

If this weren’t the case, it would be impossible to advance a field beyond what can be done from scratch in a lifetime.
Rather, it is an ongoing challenge for science to improve didactics, programmes, etc. to bring new people to the forefront of a field within a reasonable amount of time, such that they can advance this front – standing on the shoulders of giants.
As a sidenote, take a look at some really old mathematics textbooks to feel how didactics has marched on and how important that is.

That being said, trying to do some things from scratch at times can be a helpful exercise that yields a deeper understanding of the matter and trains your field-specific problem-solving skills.
But mind that this should be only as an exercise – you do not want to do this all the time.

Finally note that (at least in the fields I am familiar with), most theoretical research is not about coming up with a new theory, new axioms, or similar, but working out the consequences of existing theories.
If you can come up with a new, useful theory, that’s great, but chances are that this will only happen after you gained a considerable amount of experience with the existing theories and doing theoretical work in general.

As a complete sidenote: Doing a PhD without a supervisor experienced in your respective subject is usually not a good idea.