The gemara says that we don't accept a korban from a mumar for avoda zara or a mumar who violates shabbos publicly, but we do accept from a mumar who violates another aveira. From this the poskim derive that one who violates shabbos publicly has a reduced status in halacha, and for many things is considered like a goy. His shechita is passul, he can't be counted as a minyan, one cannot cook for him on yom tov.... Rashi explains:
דהעובד עבודת כוכבים כופר בהקב"ה, והמחלל שבת כופר במעשיו ומעיד שקר שלא שבת הקב"ה במעשה בראשית
Rashi understands that the violation of shabbos is more fundamental than mila, kashrus and other issurim in the Torah because it is a denial of G-d creating the world. However, Rashi doesn't explain why this is only true when one publicly violates the shabbos, but is not true when one privately violates the shabbos.
R. Moshe (Igros Moshe o.c. 1:33) explains the distinction and with this explains why many people nowadays wouldn't have this status. When one violates shabbos privately we consider this to be לתאבון, but when one violates shabbos publictly he is doing a very open act of rejecting Hashem as the creator who rested on shabbos. Since we aren't able to know what he is thinking, we have to look at his actions to make that determination. A public violation of shabbos indicates a rejection of the creator. Therefore, this distinction only applies when one recognizes the seriousness of shabbos and what it represents, yet makes a conscious decision to violate it publicly. But, when one has no appreciation of shabbos at all, their violation isn't a denial of Hashem and therefore it is tantamount to a private violation that we attribute to his yetzer horah getting the better of him. The Sridei Eish (R. Yechiel Yacov Weinberg) refers to this teshuva of R. Moshe and writes that he agrees with the logic.
Based on this approach many shabbos violators nowadays would have the status of a regular מומר לתאבון even if they violate shabbos publicly. However, the leniency wouldn't extend to those who grew up religious with an appreciation for shabbos and later chose to reject it.
Most non-religious Jews would have the status of מומר לתאבון, as opposed to מומר להכעיס or מומר לחלל שבת בפרהסיאץ. But, they don't have all the advantages of a mumar l'tai'avon. When the gemara speaks of a מומר לתאבון, it speaks about someone who makes decisions before he does aveiros. When he has nothing to gain and no benefit in violating, he will always decide to keep the mitzvos i.e. purchase kosher food. It is only when it is too expensive, too hard to obtain, or doesn't taste as good that he will decide to eat non-kosher. It is for this reason that when we can ascertain that he has nothing to gain by doing something wrong, he will do it right. But, for the most part, nowadays people who aren't religious, don't make any decisions at all. Meaning, they categorize themselves as "I don't keep kosher" or "I am reformed so I drive on Shabbos". Since they don't make any decisions or make any attempt to do what is right, they wouldn't be trusted to do something properly, even when there would be no gain to do it improperly. Even Rava who says that we don't say כיון דדש בה כהתירה דמי ליה - these people are כהתירא דמי ליה, where we no longer have the assumption of לא שביק התירא ואכיל איסורא.