All physical theories are fine-tuneable to some extent. The Standard Model of physics, our best tested theory, has 25 physical constants that need be put by hand. That doesn't make the theory able to predict "whatever you want". It means that you have to perform at least 25 independent physical experiments to fix those free parameters and then your theory is set and can be used to make predictions.
The reason theoretical physicist pay attention to String Theory is because it's the only consistent theory of Quantum Gravity we know of. That doesn't make it the correct one for our universe, but is the only one that we know. And since physicist have been trying to crack the problem for almost 100 years now, it makes sense that some attention is paid to it, at least in the hopes that we learn how to build other theories of Quantum Gravity.
Like you said, people take the Standard Model seriously not because they like it (in fact, there is a level of embarrassment that physicists seem to have on this because the theory is so ugly), but because it has been experimentally verified within its regime. The fine-tuned constants have been experimentally derived.
So you're right, fine-tuning by itself is not necessarily a problem ... if it's experimentally tested. If you don't have experimental guidance, what else is left? At that point, you're a mathematician working in abstract worlds.
>The reason theoretical physicist pay attention to String Theory is because it's the only consistent theory of Quantum Gravity we know o
Another String Theory wart ... the theory fits more naturally in a universe with a negative cosmological constant - which is not our universe.
To be clear, I'm not making an argument that String Theory should be abandoned. I'm not physicist or a mathematician so I don't have the expertise to make that kind of judgement, although there are credible physicists who do make this argument. But at at least part of the popularity of String Theory entails an interesting socio-cultural phenomena that should be studied and analyzed by philosophers of science.
Disclaimer: I have no horse in this race, I studied Theoretical Physics but left academia for the industry many years ago. My bills won't be paid whatever side of the discussion gets funded.
1. In its canonical formulation we don't know if Loop Quantum Gravity produces the right classical limit (i.e. General Relativity). In its spin-foam formulation we can get the right classical limit but we don't know if the formulation is mathematically consistent.
2. The semi-classical results that LQG obtains, namely Black Hole entropy, is very suspect. The semi-classical result is pretty much cooked in (at least it was years ago when I was studying these things, maybe they have refined the way the calculate it recently) and the logarithmic quantum corrections are wrong (see: https://arxiv.org/pdf/1205.0971.pdf).
3. Furthermore, it's questionable how can you come up with a consistent theory of pure-Quantum Gravity that doesn't include all other forces and matter since gravity interacts with everything (that is its most defining characteristic) and in Quantum Mechanics you must include everything that can happen otherwise the diagrams would need to conspire in very peculiar ways to produce consistent results.
TL;DR: The viability of LQG as theory of Quantum Gravity has always been overstated to the general public. By the classical journalistic trick of giving equal voice to the two sides of the discussion; the reader, with not much else to judge by, has to assume equal plausibility for both sides of the argument.
With a bit more careful inspection you would notice that the number of experts paying attention to one side or the other is radically different. So either the entire High-Energy physics community has fallen in some collective delirium and "big physics" conspires to suppress LQG or maybe LQG doesn't deserve as much attention as the general public has been led to believe.
The reason theoretical physicist pay attention to String Theory is because it's the only consistent theory of Quantum Gravity we know of. That doesn't make it the correct one for our universe, but is the only one that we know. And since physicist have been trying to crack the problem for almost 100 years now, it makes sense that some attention is paid to it, at least in the hopes that we learn how to build other theories of Quantum Gravity.