I think you are wrong; for example the Standard model has 26 (or 25?) fundamental constants that can be made dimensionless, so they are not dependent on the choice of units. Also, the masses of fundamental particles are connected to the strength of the coupling of the particle's field to the Higgs field, and those coupling strengths are fundamental constants.
Not all the parameters of the standard model can be made dimensionless, and for most others their dimensionlessness is illusory.
When you have 10 absolute masses, you can also represent them by one absolute mass and 9 dimensionless mass ratios. Then you can claim that your model has 9 dimensionless parameters, but this changes nothing, because in any formula that relates different physical quantities you must multiply the mass ratios by an absolute mass.
There are cases when using dimensionless mass ratios does not bring any advantage, but there are also cases when using mass ratios can improve the accuracy. For very small things, like particles, nuclei, atoms or molecules, it is possible to measure with much higher accuracy the ratio between their mass and the mass of an electron, than their absolute mass.
So in particle/nuclear/atomic physics it is usually the best to express all masses as dimensionless ratios vs. the mass of an electron and to multiply them with the absolute mass of an electron only where necessary. There are methods to measure with high accuracy the absolute mass of an electron, based on measuring the Rydberg constant and the constant of the fine structure.
Most parameters of the standard model are dimensionless because they are ratios that eventually need to be multiplied with an absolute value for the computation of practical results.
In many research papers about the standard model they get away with using only "dimensionless" parameters because they take advantage of the fact that they cannot continue the computations far enough to obtain practical results, so they use some ad hoc system of units that is not clearly defined and which is disconnected from the remainder of the physics.
> When you have 10 absolute masses, you can also represent them by one absolute mass and 9 dimensionless mass ratios. Then you can claim that your model has 9 dimensionless parameters, but this changes nothing, because in any formula that relates different physical quantities you must multiply the mass ratios by an absolute mass.
What? So, what is the mechanism in nature that produces those 9 ratios? If you haven't got one, you haven't got a theory; you've got to admit that those are experimentally confirmed fundamentals.
So, I interpret what you're saying like this: you have got a pet theory where the mass ratios are calculable from some other, more fundamental assumptions, but (charitably: because the theory is numerically too hard, or you lack compute resources) you don't have the actual means to calculate them.
Otherwise, I scratch my head. If you don't manage to enlighten me, I deem you a crank.
I did not say anything about any mechanism in nature.
This is just a mathematical manipulation of the numeric values. If you have the absolute masses m1, m2 and m3, you can store instead of those 3 numbers other 3 numbers, m1, m2/m1 and m3/m1. This stores the same information, because whenever you need e.g. m3, you can recover it by multiplying m3/m1 with m1.
You can do the same thing with one hundred absolute masses, replacing them with a single absolute mass together with 99 dimensionless mass ratios.
As I have already said, sometimes there are good reasons to do so, e.g. in particle/nuclear/atomic physics you can increase the accuracy of computations if only a single absolute mass is used, the mass of the electron, for which there are accurate measurement methods, while all the other masses are replaced with dimensionless ratios between the corresponding absolute masses and the mass of the electron, because such mass ratios can be measured accurately based on the movements in combined electric and magnetic fields.
This includes the standard model, where the absolute mass of the electron is necessarily one of the parameters of the model, but all the other masses can be replaced with dimensionless mass ratios, e.g. the mass ratios between the muon mass and the electron mass or between the tauon mass and the electron mass.
Similarly, any parameter used to characterize the strength of the electromagnetic interaction is not dimensionless (in any system of units where the elementary charge is one of the base units, and such systems are better than the alternatives), and one such parameter must be included in the standard model. The parameters that characterize the weak interaction are also not dimensionless, but they can be converted to dimensionless parameters in a similar way with the masses, by using ratios (vs. the electromagnetic interaction) that are incorporated in the mixing angles parameters.
The end result is that the standard model is expressed using a large number of dimensionless parameters together with only a few parameters that have dimensions, because this representation is more convenient. Nevertheless, there are alternative representations where most parameters have dimensions, by rewriting the model equations in different forms, so there is nothing essential about having dimensions or not.
