Is the right hand rule a trick to avoid tensors?
Angular momentum is most naturally a skew two-index tensor $L^{ab} = x^ap^b-x^bp^a$. Indeed, in higher dimensions, the two-index language is essential. In 3-d the $SO(3)$ invariance of the Levi-Civita symbol $\epsilon^{abc}$ gives us the option of converting $L^{ab}$ to index object to a one-index (cartesian vector) object $L^a= \epsilon^{abc} L^{bc}$ - but we have to make a choice of sign when we do this. The right-hand rule is just a convention, as are right-handed co-ordinate systems. You can do otherwise (and at least one classic book on mechanics uses a left-handed coordinate system) but then you will have difficulty communicating with other people...
Not quite an answer to the question asked, but a clarification of one of the ideas you used to build the question.
As I have noted elsewhere right- (or left-)hand rules tend to appear in pairs when computing physical behaviors, so the resulting physics is reflection invariant.
Yes, you use a right hand rule to compute the angular momentum of a spinning wheel (a quantity, not a behavior), but you use another right-hand rule to find the direction of a gyroscope's precession (a behavior) from that angular momentum.
Likewise you use a right-hand rule to find the direction of a magnetic filed created by a solenoid (a quantity) but another right-hand rule to find the direction of a ion's curvature in that field (a behavior) from the field.
In both cases the behavior is unchanged if you use left-hand rules instead of right-hand rules. (Though the convention for direction of angular momentum or magnetic field is reversed.)
In other words, the presence of handed rules in physics doesn't automatically make the behaviors of systems described parity odd (or worse fail to have a well defined parity).