I'm not sure about that. Some ideologies are certainly more conducive to morality than others, but for me, the question of whether something is moral, good, or right, always revolved around the question:
If everyone did this, would the society as a whole work well?
And you can see right there that stealing, killing, and hurting others won't make things work. Basically it's the notion that if you don't want certain things to happen to you, you shouldn't do them to others, because for the society to function well, the "goodness" has to be reciprocal. That's, imo, a solid basis for morality without requiring anything religious. You just need some understanding and ability to abandon a completely egocentric view and see things from others' perspective.
This is the rationalist, Kantian, and partly Sam Harris-like stance on it. But I'm not so sure it can work like that, at least not without some additional (and murky) part played by feelings, conscience, and unconscious beliefs. It's just not as straight-forward as the Kantian/Harris types make it seem like - that all you need is some rational thinking and off you go.
The problem is that while you can think about the greater benefit for society, you are still strongly tempted to act immorally for your own benefit. If you know you'll get away with it, if you know nobody will ever find out, and you know the gains are large - then it's kind of hard to engage in Kantian gymnastics in such a situation, isn't it? Why not just charge a little more than you deserve, why not take subtle advantage of someone, why not spend the evening binge-watching on Netflix? Why not leave the dishes to your spouse because you are in no mood washing dishes, why not tell a little lie to make a problem go away instead of fixing it? Why indeed?
This is not to say that people can't do the right thing without religion, but still, there needs to be something more IMO, at least in my experience. Because doing the wrong thing is always the easiest option. And it's so, so damn hard to do the right thing in many situations that we need everything we can muster to fight our bad tendencies. Like a super-powerful conviction that every single little thing you do matters and reverberates throughout the universe. Like the conviction that you will be held accountable in the next life for your actions. Like the feeling that there are powerful forces beyond your control that represent a higher order that you violate at your own peril. Things like that.
In the book "Paul and the Stoics", Troels Engberg-Pedersen lays bare the common understanding of both Paul and the Stoics that your ethical stance must completely fill your life, in everything you do, all the time. That way, transformation can happen. With Paul, it's "being in Christ", with the Stoics, it's more about an all-encompassing rationality, though it's important to define these words properly. I think the point is: you can't just think about this or that ethical conundrum when you encounter it, using Kantian philosophy or whatever. No,
you must strive towards the Good and the Beautiful in every second of your life, with every breath you take. But since we humans forget ourselves all the time, and it's so difficult to boot, we cannot do this without a motivation and dedication of supreme power.
I agree with you that simple, authoritarian religious dogma doesn't cut it, and that rationality and critical thinking play a big role in all of this. But again, I think you need more than that - and that "more" does point towards a higher realm of some sort, however you choose to think about it. It could also be a connectedness via our higher centers to the realm of "objective morality", something we can sense if we pay attention to it and actively work towards overcoming our base natures. And in a sense, thinking about such things rationally CAN be the key. But some primitive utalitarian "let's maximize happiness" doctrine à la Sam Harris doesn't even begin to address the issue.
A final point: if you are an atheist, you tend to elevate your own thinking, your own rationality to the level of God. You think you can know right from wrong just by thinking things through - as if you could foresee what's best for you, for society, for the world. But things always turn out to be much more complex than we think. And how many "rational", well-intentioned ideas have led to so much suffering!
I think this fact, that we are not God and cannot "calculate" or even know what's "good for the world", means we need to "play ball" with God/the Universe, in the sense that we need to pay close attention to our interactions with the cosmos and how it responds. There is constant communication going on through the medium of our lives unfolding. I think learning to understand this "language" is more important than thinking about our decisions in terms of "if everyone did this, would it be good for society"? We can't know, and besides, it's kind of missing the point: we are interacting with the Cosmos individually. What's good for us may be bad for someone else, and vice-versa.
Sorry for the long post - I think it's an interesting question and not one that can be easily settled.