Amnesia is very important in anesthesiology
Indeed, this is a concern in the anesthesiology community as well. I recall a friend and former colleague of mine was involved in a project including this paper (in which I had no involvement):
Raz, A., Eimerl, D., Zaidel, A., Bergman, H., & Israel, Z. (2010). Propofol decreases neuronal population spiking activity in the subthalamic nucleus of Parkinsonian patients. Anesthesia & Analgesia, 111(5), 1285-1289.
...in which they attempted to characterize brain activity under anesthesia that is normally measured in awake subjects while placing deep brain stimulation electrodes for patients with Parkinson's disease. Usually this procedure, like those you mentioned, requires patients to be awake because the position of the electrodes depends on brain activity. General anesthesia disrupts this brain activity, so it hasn't typically been used.
This just a case study, though: it's a general area of interest in anesthesiology to extend general anesthesia to cases where it is normally avoided in order to improve patient comfort.
So yes, this is definitely an area of concern and consideration. However, since this is a concern and yet there isn't an ideal agent in use, it's fairly easy to conclude that the answer to:
is there a (simple & magical) way of inducing anterograde amnesia without the mental "fog" for trauma reduction in surgery?
is unfortunately no, otherwise it would be in use all the time! However, for at least some procedures, benzodiazepines can help.
Benzodiazepines promote amnesia and enhance anesthesia
As you say, anterograde amnesia is a beneficial "side effect" of general anesthesia, but not only that: it is also an intentional effect. Patients expect to not remember their surgeries as well as not be aware in the moment (Rowley et al 2017 is a recent survey of patient attitudes towards anesthesia). Benzodiazepines are sedatives that are often given along with general anesthesia (midazolam, in particular) to both reduce the dose of anesthetic necessary to obtain a desired hyponotic effect and also to improve per-operative amnesia.
Midazolam is an effective amnestic in moderate doses, and it acts quite quickly (see for example Dundee & Wilson, 1980), helping reduce possibility for traumatic experiences even in the initial preparations for surgery.
Benzodiazepines as an alternative to general anesthesia
Some procedures are done with only benzodiazepines as "conscious sedation" (often with addition of a narcotic or local anesthetic since benzodiazepines do not provide analgesia), both for the anxiolytic and amnestic effects as well as safety compared to general anesthesia (subanesthetic doses of general anesthetic agents like propofol may be used alternatively or in addition in some cases). Examples include dental procedures, colonoscopies, and endoscopies. Often these procedures can be performed with limited patient memory, especially after 24 hours.
Benzodiazepines may not be appropriate for all patients or all procedures, of course, and they are certainly sedating even if patients are conscious. However, there is no safe alternative for inducing amnesia without full loss-of-consciousness.
Dundee, J. W., & Wilson, D. B. (1980). Amnesic action of midazolam. Anaesthesia, 35(5), 459-461.
Rowley, P., Boncyk, C., Gaskell, A., Absalom, A., Bonhomme, V., Coburn, M., ... & Sanders, R. D. (2017). What do people expect of general anaesthesia?.