"It is difficult to understand why someone who wants to be a Jew and makes a serious effort at it would not be considered one"
The person (lady) in the article does not seem to show this. The opposite is true. It is either all or nothing, someone not keeping the Shabb-ath, not observing family purity and calls themselves a "convert" is not valid, as with the lady in this case. And about your question about those people born of Jewish parents who do not observe these things- on a level you are right, becuase a Jew who doesn't observe the Shabb-t is like a goy, but the only difference is that they can become "fully Jewish" (Quote on quote) once they make tishuva, while a non-Jew has to go through a conversion process. -(just so their is no confussion- the non-observent Jew is Halahically or legally Jewish, but on a level he or she does cut themselves off when they break very important things like the Shabb-t and family purity).