I'll try to explain the permissibility of including any Jew as part of the minyan at all times, summarizing answers I got on this discussion from 3 notable Orthodox rabbis.
Rabbi #1 had a small shul and on Shabbat many of them drove. A number of the walkers asked him why he allows this. He answered that he does not explicitly tell them to drive to shul on Shabbat. But, if he excluded them, inevitably, they would drive to the Reform shul in the next neighborhood. There, they would be less likely to be exposed to Shomrei Shabbat & Shomrei Mitzvot people. Thus, in a sense, the rabbi was concerned about kiruv. He felt that by encouraging them to come to his shul, when they see the beauty of the shul and Shabbat, hopefully, on their own, they would become Shomer Shabbat - of course with some assistance from him.
The other two rabbis concurred with his viewpoint, as they had similar problems in their own shuls.
In Rabbi #1's case, because he was in a small community, he stated that if the drivers didn't come on Shabbat, there would be no minyan at all, at in a sense, that would penalize him as well as the other walkers / Shomrei Shabbat people who attend shul. And, then, what would be the point of having his shul in the first place if no one came? I.e. - a firm "No" would have repercussions as he feared that to find SOME minyan on Shabbat, the Shomrei Shabbat people would daven in the Conservative or Reform shul. So, should his shul always remain empty on Shabbat while the other shuls "pick up the slack"?
The other 2 rabbis agreed with the 1st rabbi's view. As for the rest of the week, ALL 3 shuls have had a problem with getting a minyan, even now. All 3 rabbis are happy when any Jew davens with a minyan. He cared enough to daven and he helped others daven with a minyan, esp. those who have to say Kaddish. For many of them saying Kaddish, alone is what makes them attend a minyan in the first place! All 3 rabbis have stated that in the majority of cases, when those that said Kaddish were not Shomer Shabbat, it was directly because of their 11-month experience of attending the minyan daily that many of them became Shomer Shabbat - something that almost certainly would NOT have happened on its own had the rabbi excluded them from the minyan.
And, one rabbi pointed out that while Gentiles are not included in a minyan, they are invited to come to a shul and pray with us, if they wish. So, we should welcome a Gentile, but exclude a non-Shomer Shabbat Jew? (By excluding, he didn't mean that he would expel him from the shul. But he meant that they would exclude themselves because we made them feel unwelcome.)
In brief, I have no direct halachic proof from any of these rabbis. But, I think that their view of kiruv and their insight into solidifying an often weak community by finding some leniency is notable and encouraging.
If you'd like me to inquire further about any specific aspect, inform me as I am very much in contact with 2 of the 3 rabbis.