In other words, claiming that rather than the universe being fine tuned, it just fit the existing conditions out of necessity.
No, the puddle argument is a response to the claim that the environment is particularly suited to humans, and it asserts that, rather than the environment being suited to humans, humans are suited to the environment.
The most common response from eg atheists is comparing the situation to water in a puddle remarking on how the hole in the ground is exactly the right shape to hold it.
Really? When I Googled "puddle analogy", I got these results:
Explaining the anthropic principle.
Stand to Reason article "Why the Puddle Analogy Fails against Fine-Tuning"—A Christian apologist explaining why the analogy doesn't defeat fine-tuning.
Reasons to Believe article "Does the Puddle Analogy Explain Cosmic Fine-Tuning?"—ditto
Quora question "What does Douglas Adams’ Puddle Analogy mean?"—Question about the analogy
The Truth Will Make You Mad article "The Puzzle of Existence and a Puddle of Doubt"—Christian apologist attacking Douglas Adams, claiming that he misrepresents the fine-tuning argument.
Now, here are the results for "fine-tuning rebuttal":
Why Evolution Is True article "Sean Carroll debunks the “fine-tuning” argument for God"—No mention of "puddle" in the article, although someone brings it up in the comments.
Stanford Encyclopedia of Philosophy article "Fine-Tuning"—No mention of "puddle".
Reasons to Believe article "Fine-Tuning: Responding to a Common Objection"—One of the Christian apologists from above.
The Atheist Voice video "A Rebuttal to the Fine-Tuning Argument"—Does mention puddle, but only once, in passing.
Rashomon Effect video interview "Intelligently Design, Rebuttal To Fine Tuning Argument | Richard Carrier PhD"—No mention of "puddle".
My search for "puddle analogy" was dominated by Christians claiming that it was an argument made by atheists, without any results (at least not in the top 5) of atheists actually making the argument, and my search for rebuttals of the fine-tuning argument was dominated by non-puddle arguments, with a few off-hand mentions of the puddle argument.
Now, let's look at Douglas Adams' original puddle analogy:
Where does the idea of God come from? [...] Now imagine an early man surveying his surroundings at the end of a happy day's tool making. He looks around and he sees a world which pleases him mightily: behind him are mountains with caves in - mountains are great because you can go and hide in the caves and you are out of the rain and the bears can't get you; in front of him there's the forest - it's got nuts and berries and delicious food; there's a stream going by, which is full of water - water's delicious to drink, you can float your boats in it and do all sorts of stuff with it; here's cousin Ug and he's caught a mammoth - mammoth's are great, you can eat them, you can wear their coats, you can use their bones to create weapons to catch other mammoths. I mean this is a great world, it's fantastic. But our early man has a moment to reflect and he thinks to himself, 'well, this is an interesting world that I find myself in' [...] early man is thinking, 'This world fits me very well. Here are all these things that support me and feed me and look after me; yes, this world fits me nicely' and he reaches the inescapable conclusion that whoever made it, made it for him.
This is rather as if you imagine a puddle waking up one morning and thinking, 'This is an interesting world I find myself in - an interesting hole I find myself in - fits me rather neatly, doesn't it? In fact it fits me staggeringly well, must have been made to have me in it!'
There is absolutely no mention of the fine-tuning argument here. Adams is not responding to the argument that the conditions are the only ones that make life possible, he is responding to what he imagines the belief of early humans to be, namely that the conditions of humans were particularly suited to humans. One can argue that the label "fine-tuning" sort of fits this line of thinking, but it is a particular sort of fine-tuning, and is not the argument that people usually use "fine-tuning" to refer to.
So, all of that is a frame challenge. What about your actual question? Well, the Christian apologist response to the puddle argument is quite simple: they present an argument that it was never presented as a rebuttal to, point out that it's not a rebuttal to that argument, and then smugly pat themselves on the back about how smart they are. In Adams' analogy, the puddle corresponds to humans, the shape of the puddle corresponds to humans' evolutionary adaptations to their environment, and the hole corresponds to that environment. Humans of course couldn't evolve to fit a universe that didn't allow life in the first place, so of course the puddle analogy doesn't address the claim that the universe is uniquely capable of supporting life. There are plenty of other explanations for how the fine-tuning argument is incoherent, fallacious nonsense, such as in Rational wiki's article "Argument from fine tuning" (again, no mentions of puddles).
The fine-tuning argument implicitly (or, more rarely, explicitly) relies on the intuition that is formalized in Bayesian analysis. But for us to do Bayesian analysis on this question, we would have to take the total number of universes with fine-tuning and a God and divide that by the number of universes with fine-tuning with or without a God. Besides the question of what possible meaning this could have, and how we could figure out what number of universes have fine-tuning and a God (and how asserting that this number is large isn't begging the question), the argument pivots crucially on asserting that the number of universes with fine-tuning and no God is minuscule, at least compared to the number with fine-tuning and a God. This is a positive claim, and the burden of proof is squarely on the apologist. And yet when atheists dispute this claim, for instance by pointing out that there are hypotheses under which such a large multitude of universes are created with different constants that it is inevitable that some of them will have "fine-tuning", this is dismissed by apologists as a "cop-out" with "no evidence" and purely "philosophical" "grasping at straws", as if it's the atheists that somehow have the burden of proof.