I am considering a possibly risky route into graduate school. Namely, auditing classes at the university I would like to attend and then reapplying after a semester or two. This way I can demonstrate my abilities directly.
I cannot speak for Physics, but I can speak generally. Generally, auditing courses is unlikely to get you noticed in any way and is unlikely to generate the sort of information that the admissions committee will value. I cannot assume that you will be able to develop the connections that you desire because I doubt such opportunities will be available via auditing classes.
In my field (though perhaps not Physics I warn) a way forward with a poor undergraduate result but a desire to do a US-based PhD is to do a one-year master's in the UK or Europe, and to do extremely well in that programme. Via this route you show that you are able to complete very high level coursework and, further, are able to produce actual research (in the form of a master's dissertation). These are much better indicators of PhD success than audited classes or GRE scores. Also, if you have, say, a UK distinction in your master's, in your PhD statement of purpose you can address your poor undergraduate performance head on: say you were dealing with personal circumstances but your demonstrable success in the higher degree programme evidences your abilities to succeed at real research.
In other words, you are giving them real evidence of your commitment and abilities, while auditing doesn't do that.
What you are suggesting has a few, significant, pitfalls.
(1) You may find it hard to sustain the effort, especially when not doing it for credit. Note, this is very common issue. Look at the statistics on it or on similar situations (people doing extension classes online).
(2) It may not be really credited. After all, you won't get a grade out of it.
(3) Teacher may not give you as much help and/or you will be reluctant to ask for it, as not being a paying student.
(4) Kinda limits you mostly to the one school where you do the courses. But you need to cast a wide net.
(5) Low value if stepping stone doesn't work. Whereas a masters or "less prestigious" Ph.D. has value even if you don't end up at Harvard playing Bessel functions with Lisa Randall.
I would go ahead and apply to many regular schools. Prestigious and not prestigious. For one thing hard science grad school can be quite different than undergrad in terms of the demand (they need students). You may well find that some school is willing to take a chance on somebody who is "smart and lazy". (Sorry, but that is what the quick analysis of high test scores and low grades will determine. If it was personal troubles, you're right to keep those out of the application.)
Also, prestige is not the end-all, be-all. But if you do go to a less prestigious one, I would at least try to go to a large state school (Big 10 or UT or the like). At least you are part of the core of R1 world then.
Finally, I had a buddy who leveled up in physics. Going from a so so place (but in a Ph.D. program, not auditing) to JHU. They were happy to get him. He's a smart cookie.