Yes, but the thing is I never see facts on these cases, which is what leads me to believe its horsecrap. Its as simple as that really.
What do you mean by "I never see facts on these cases". The Beeb has just given you a bunch of facts. Are you saying that the Beeb is misrepresenting the truth?

Hypnosis as an anaesthetic alternative is well documented in medical journals, and confirmed by many studies and tests with control groups.
Like this one:
The Effectiveness of Adjunctive Hypnosis with Surgical Patients: A Meta-Analysis
Hypnosis is a nonpharmacologic means for managing adverse surgical side effects. Typically, reviews of the hypnosis literature have been narrative in nature, focused on specific outcome domains (e.g., patients’ self-reported pain), and rarely address the impact of different modes of the hypnosis administration. Therefore, it is important to take a quantitative approach to assessing the beneficial impact of adjunctive hypnosis for surgical patients, as well as to examine whether the beneficial impact of hypnosis goes beyond patients’ pain and method of the administration.
We conducted meta-analyses of published controlled studies (n = 20) that used hypnosis with surgical patients to determine: 1) overall, whether hypnosis has a significant beneficial impact, 2) whether there are outcomes for which hypnosis is relatively more effective, and 3) whether the method of hypnotic induction (live versus audiotape) affects hypnosis efficacy. Our results revealed a significant effect size (D = 1.20), indicating that surgical patients in hypnosis treatment groups had better outcomes than 89% of patients in control groups.
No significant differences were found between clinical outcome categories or between methods of the induction of hypnosis. These results support the position that hypnosis is an effective adjunctive procedure for a wide variety of surgical patients.
IMPLICATIONS: A meta-analytical review of studies using hypnosis with surgical patients was performed to determine the effectiveness of the procedure. The results indicated that patients in hypnosis treatment groups had better clinical outcomes than 89% of patients in control groups.
These data strongly support the use of hypnosis with surgical patients.
Source.
Examples could be multiplied. This is not "fake"; it is a real phenomenon, proven by medical science. Hypnosis is not a trick; it's just a mental discipline that anyone can learn.