Hi, a bit of info, an Oral/Maxillo-facial surgeon is firstly a dentist who then usually does a shortened medical degree as well. After several years further training they will work as a consultant in a hospital. Their work will involve complex surgery due to oral cancer, jaw problems and difficult surgery of teeth, this can include difficult wisdom teeth. Some of their work will be done with local anaesthetics but most, by it's nature will need General Anasthesia. This, thankfully, is now only done in hospitals but there have to be compelling reasons for using it for routine dental work.
An alternative to General Aneathesia is sedation, usually combined with local anaesthetic which can give the best of both worlds. However thare has been a move away from single-operator work like this and certainly General Anaesthesia is not to be undertaken lightly as both dentist/oral surgeon and anaesthetist are vying for position in the mouth, one is trying to get the operation done while the other is trying to keep the patient alive.
Your normal dentist will be far, far more experienced at local anaesthetics and removing routine wisdom teeth than an oral surgeon, indeed an oral surgeon would not dream of doing ordinary wisdom teeth unless it were privately.
Any wisdom teeth work will involve some swelling of the face and indeed bruising down the neck for a week or two, That is normal, inevitable and does NOT mean something has gone wrong.
Pain relief afterwards is often necessary as the local anaethetic wears off but different analgesics suit different people. Valium for instance does not relieve pain, it is a sedative, all of them have side effects and just because you may get some of these does not mean they are useless or wrongly given or you have an allergy or some other more serious but undisclosed fatal disease
So to sum up, do be careful about the advice you get here, although really well meant, with one exception most of it is anecdotal and scaremongering. Please listen to the experts who actually know.