Dimensional analysis doesn't need to be perfect, it just needs to rule out some options. You can convert all units to mass and still get useful results, but the more diverse you can keep the units the better.
I suppose you could even go through the relevant laws of physics and only identify the units for quantities that need to be added together (or to make the quantities inside more complex functions unit-less).
In the case of torque and energy you basically run into the problem that torque times angle is an amount of work and angle is traditionally unit-less. You could however give angle a unit by introducing some 'unnecessary' conversion factor like 'degrees per radian', which is only used for trigonometry. In that case you could reformulate torque as 'energy per angle'.
I suppose you could even go through the relevant laws of physics and only identify the units for quantities that need to be added together (or to make the quantities inside more complex functions unit-less).
In the case of torque and energy you basically run into the problem that torque times angle is an amount of work and angle is traditionally unit-less. You could however give angle a unit by introducing some 'unnecessary' conversion factor like 'degrees per radian', which is only used for trigonometry. In that case you could reformulate torque as 'energy per angle'.