Yes. There are simply many answers to the fine tuner argument. One is to deny teleology. Others are as you have outlined.
But we cannot assume that the universe is devoid of a purpose just as we cannot assume that it has one.
The fine tuner argument, for a variety of reasons, moves us no closer to God. It does uncover prejudices quite nicely, though.
Ultimately, we seem to be left with two ways of understanding (finding meaning within) the world. One concerns what *is* in a physical sense -- the empirically, naturally grounded scientific method. While we can discuss God to some degree with that method, it doesn't help much, especially if we already define God as supernatural. I guess a good way of thinking about this is that it is the rational/reasoning view of God and the universe and it is necessarily limited/stunted. Think language dominant hemisphere and story-time. This method of enquiry is good at uncovering underlying assumptions, as are often carried into discussions about God -- discovering the assumptions beneath the fine tuner argument is a very good example of this. But it can never tell us how God works.
The other is, for want of a better term, symbolic or mythological and tied to our existential concerns. Think emotion/non-language dominant hemisphere/non-story-telling/holistic. We feel meaning in certain experiences and/or with certain types of accounts.
One of the problems we run into is when we confuse the later with the former. Religions largely hang out in the non-dominant hemisphere; but we don't speak with that hemisphere, we don't tell stories. When the rational/story-telling/language-dominant hemisphere tries to make sense of what the non-rational/holistic hemisphere feels, we end up with what we call religion (amongst other things, sometimes including scientific discoveries or great stories). And it probably always gets it wrong, because it always presents the 'information' in a mediated form. Fundies confuse the mythic/existential stories of their religion with physical reality; and that creates a world of hurt.
I don't see what's wrong with calling the emotive/holistic side non-rational. In a way, that is a compliment. It isn't apples. It's oranges. We are constructed with both cognitive styles. I think the important thing is not to confuse the one with the other -- one tells us about the natural world and the other tells us about our inner world. Whether or not there is some connection between our inner and the outer world is a whole other issue that we cannot use our rational mind to answer well enough.
Maybe yes, maybe no. Who knows?