I think what the SO answer means is to use a custom script that will execute some commands. This seems the simplest.
You can also use a make file for that with a specific rule to build the readme. See as example the one use in knitr 
This seems a good use of make file. You could have a build-readme rule that is executed each time you build the vignette. I let you dig about makefile if you are not familiar.
Another solution would be to make advance use to Rmarkdown features. You can setup everything to reuse chunk in README and vignette, like showcased in this blog post and from this
example
Also, I have never done it but I think this is possible to use ref.label and knitr::all_labels() to generate the README from the vignette directly... Not sure how it would behave on CRAN though. 
Makefile solution could be the more common and standard.
If you have a
on github to play with, I can help setup all this.