There are bound to be philosophers who have tackled these questions in detail. I can't make any recommendations here as I lack the requisite knowledge, but I do have a proposition that starts with the following question:
"What sort of knowledge and information do we rely on independently of the scientific method?"

We can't come up with an exhaustive answer, but what immediately comes to mind would be social skills, gut feelings, "common sense", familiarity with discursive and rhetorical styles, any understanding of the world that has no control group and can probably never benefit from one. Of course, there are plenty of counters to any of these elements. You could say that some people deploy the scientific method in some way with social skills and so on. Or point out the LessWrong community's daily use of Bayes' theorem as they live their lives. Overall, the vast majority of our daily decisions are non-scientific and driven by feelings and intuitions, even if you can apply a top-down conscious modification of your lifestyle to meet goals, better your conditions over the long term etc.

The main conclusion we are left with is that there is a plurality of understanding across human beings that is not quantifiable but decently predictable across an undefined range of subjects. You can make a joke in a certain setting, and expect people to laugh fairly reliably (assuming you are good at this of course). You can talk to HN users and have a strong intuitive sense of what sort of discussion they might want to engage in. A person who is naturally talented at humor will not be rigorous in an explicit sense, but their ability to be funny will be rigorous in its rate of success. That bad art website you linked to (correctly) assumes that there is a sense of what bad art is and that much of the population shares that natural-mixed-with-cultural intuition. They publish the website with the expectation of the audience understanding it.

So essentially, the 'something else' we are looking for can simply be to accept that when we talk about art, we talk about a predictable understanding that is self-hosted across multiple sentient serv... uh, sorry, minds we are likely to interact with, and that can also be reliably understood from scratch. Now, you probably let out a groan at this point because we seem to be essentially back to your initial frustration: a bunch of people whose intentions aren't clear talking about equally unclear things as if they were obvious to their listeners, and being slippery when questioned.

But! the key addition here is the "understandable from scratch". You can expect any human being to understand artistic value to some extent (unlike, say, Terence Tao level math which I am confident I'll never understand), but gaining that understanding will require social and cultural engagement over a period of time on their part. The rigour we seek is something we will have to demonstrate ourselves in the intensity of our efforts to engage with art and its context.

One reason that good art is often defined by its ability to stand the test of time, is because this indicates a solid reliability in getting people across very different eras and mental worlds to engage with it successfully and be willing to 'steelman' the artists' vision.