I'm still working my way through the DWP ESA/UC assessors manual, but my impression is that the only way for a person with M.E. (with no other health conditions or disabilities) to get into the Support Group of ESA (or the UC equivalent) is if the Health Care Professional (HCP) accepts
Precisely, I was very lucky that at my ESA assessment last year I had a nurse who immediately told me she knew about M.E. and didn't need to laboriously verify what I had written on my form. She also told me that I should have been granted the home visit I asked for and was refused. This was a very different experience to previous assessments and unfortunately felt like the exception rather than the rule.
ETA: I did get put into the Support group thankfully.