Leaky gut syndrome is a real thing, but much more so in a research context than as a reliable useful clinical diagnosis, with effective treatments. Doctors are generally not going to diagnose it often, as there really are no consistent reliable tests for it or treatments for it that have been proven to work.
It is a lot more likely in the context of chronic gut problems like inflammatory bowel disease or irritable bowel syndrome, especially the diarrhoea type. So if your gut functions well and does not hurt it is less likely. The are hundreds of studies of various pro, pre and post biotics that can help, but all the studies are in general small and not close to gold standard evidence, same goes for fiber supplements. What is generally regarded as a healthy diet, or a healthy diet for the gut is likely to help, so mediterranean diet, mostly plants and not too much food overall, enough but not extremes of fiber, some fibers or resistant starches might be better than others but none are really well proven , and depending on the individual some can cause adverse effects like pain bloating or diarrhoea, especially in the context of ibs.
Almost certainly a bunch of weird gut bug , endotoxin and food chemicals leaking into your bloodstream are going to increase inflammation, and there is evidence it can contribute to auto immune disorders, what is much less proven is can it be easily diagnosed or treated in a reliable consistent way.
Before I lost weight at 145 kg, I had ulcerative colitis supposedly in remission on treatment, but with chronic daily diarrhoea, and chronic daily gut pain for over a decade. Not sure what crp was then but ESR was 44 which is fairly high in the absence of a definite cause. I have zero doubt I had leaky gut then, and that may well have contributed to developing coronary artery disease and early heart failure as well as made me feel pretty crappy in general. But that is a fairly extreme example, it would be much harder to say for sure if someone really has it going on in less extreme circumstances. With massive weight loss and a much better diet inflammatory markers and gut symptoms are much better, so I assume it is gone now.
In general a healthy diet and adequate fiber is all you can really do, maybe probiotics?