I didn’t say that using “the way of” in other sentences would be grammatically incorrect, just that it imparts a different meaning altogether. Taking the first example from the google search query that you provided, “the way of the master” refers to habits and approaches that people might adopt in an effort to emulate the way Jesus shared the gospel. The key difference here is that although one can partially embrace these Christian habits, the same cannot be said of citizenship! This should be your guide when determining which preposition to use.
The first step on the way to sanity is also the first step on the on the way to becoming sane.
The first step on the way to fitness is also the first step on the on the way to becoming fit.
the way of the Sufi is mystical
The way of the Dao is the way of least resistance.
The first step of reconciliation is often a phone call.
The first step on the way to reconciliation is likely to be a phone call.
The first step on the road to righteousness is to do unto others as they do unto you.
The first step on the road of righteousness is to do unto others as they do unto you.