Units are to calculations as Types are to code. They help prevent a large class of common errors. If your types aren't consistent, you know your code is wrong. If your units aren't consistent, you know your calculation is wrong.

Just like there can be stronger and weaker type systems, there can be stronger and weaker unit systems. Stronger unit systems catch a larger class of errors. I commonly catch errors nobody else sees because I use a stronger unit system than they do, which is amended with generics ("kg of what?"). Just like you said: a unitless quantity is weak; ppm doesn't help you catch errors very well. It could be mg/kg or µL/L, and it matters when you start canceling ratios. But you can take it further: mg<constituent>/kg<soil>. When you start throwing in soil density and molecular mass, it's very easy to come up with examples that "pass" the standard dimensional analysis but "fail" your advanced unit checker. You can look like a wizard by detecting errors in equations completely outside of your field of expertise.

Once you realize that there can be as many unit-systems as type-systems and the only criterion for a good vs. bad unit system is how well it catches errors, the purists who try to enforce their physics textbook's view of units will look really silly to you. The only problem is that you start arguing with them about whether specific gravity is really unitless or not (it's more helpful to say it's not), and they immediately decide you must be a moron because that's not what their textbook says. Oh well. You're getting the right answer, and they're not.