Let's start by admitting that most Christians in any given era buy into the idea of rewards. Christians debate the nature, extent and timing of such rewards. That being said, many Christians in any given era teach the highest standard involves becoming the kind of person who practices righteousness without thought of reward. Fuzzy thinking, biblical illiteracy, unexamined religious traditions, and human nature often combine to complicate the picture.
That being said, I see the matter as follows.
(1) Jesus did not teach that heaven is a reward. Heaven is a condition, potentially experienced at any moment and potentially eternal. Heaven is to live in right relationship with God in each instant. Heaven is the kind of life God intended for humanity.
(2) It is incredibly difficult for any of us to undertake acts of righteousness without weighing the possible benefits to ourselves. Such benefits may include the approval of others. More subtly, we may fall into the trap of praising ourselves, of basing our sense of worth on the number or quality of our righteous works. Both approaches are highly self-concious, and both push us to do good things for the sake of a reward.
(3) In the passage, Jesus teaches his followers not to let one hand know what the other is doing. The core idea seems to be that his disciples are to be unselfconcious. In such a state, they may do good things without thought of any reward. They become so immersed in life with God that they no longer pay attention to the response of others or even themselves to their good works. Ideally, they become selfless.
(4) Jesus adds a surprise: God rewards the selfless. Such people will not expect a reward. They will be surprised by it, and are apt to try to give it away to someone else. They are like those who respond (paraphrased): "When did we see you in trouble, Jesus, and help you." They do not serve for the sake of any reward but instead do so because such service is part and parcel of their identity, an identity shaped by their growing intimacy with God. If this is true, I do not think Jesus abandoned the rabbinic standard of lishmah so much as he recast it.
With all of the above in mind, I tend to look upon the Christians you describe as people who have taken a first step. God has spoken to them, calling them to himself. Bound by culture (usually many layers of culture), they hear and interpret the call in terms of the life they know. There is no shame in starting from where one is, and God loves us more than enough to come and find us where we are (that's one implication of Incarnation, by the way). First steps, though, should never be last steps. As we follow Jesus, all of us should grow in our capacity for selflessness. Selflessness, of course, is only possible for those who depend upon God for all they actually need.
Finally, I think I understand the facination of the phrase "your Father who is in secret." How could any mystic not focus on it? In the case of the passage, though, I think it wise not to read too much into matter. The essential idea is that "God knows." God knows what it is like to work quietly and unobserved, and God sees all that is done in secret, including quiet deeds of righteousness. The more deeply we move into relationship with such a God, the more we adopt God's mode of operation.