You can pretty much diagnose yourself with the great EDS tools available on EDS sites. Since most of the markers are physical things that you can measure yourself, you can be pretty accurate. If you test yourself on an EDS site and the results clearly indicate EDS, you could just show the results to one of your doctors and get them to put it in your medical records. And, if you seem to have EDS, you could just incorporate life-style measures into your life. Unfortunately, there are any good treatments I have found other that strengthening muscles to make up for lax connective tissue.
Sure. Though not having a formal diagnosis doesn't help with things like additonal medical support or getting disability pension. It can mean that it's far more difficult to get further medical testing for things like CCI etc.
Also, there are other hypermobility types; it'd be good to be able to pin these separate things down and work to each type of issue to the best of research, knowledge and ability.
YMMV of course, but I'm always going to be after as precise a dx as possible where I can. Especially as one CFS 'expert' here told me I 'probably' had POTS but 'not to bother' getting tested for anything after my ME/CFS diagnosis as it would 'likely just lead to further issues being uncovered'(??)... but since then, another Dr actually treating my POTS with off-label meds has helped me regain significant enough functionality that I can still support myself.
In a semi-related vein, I skimmed some research the other day about how crowded teeth may affect the skull-neck axis to the point of causing serious axial instability. And, as those with EDS/HSD probably know, having hypermobility/connective tissue disorder can affect teeth crowding a lot.
There are flow on effects worth considering re connective tissue, whether it's genetic or a gradual tissue break down due to things like Coxsackie and/or other cumulative effects of long term ME/CFS.