I am an Anesthesiologist in the US. Yes, in the US most of those tubes are put in by the Anesthesiologist.
People tend to think of Anesthesiologists as being experts in giving drugs to make someone sleep. That is actually easy to do. The problem is keeping them breathing and the ABCs:
Airway, Breathing, Circulation. People don't usually die from drug overdoses, they die from lack of oxygen to the brain because they stop breathing. THE experts in keeping the airway open and keeping someone breathing are Anesthesiologists.
I could give a person a synthetic narcotic that is 100x as potent as Heroin and give them a 100 fold overdose and keeping them alive would be easy for me: just breathe for the patient until the drug wears off.
Developing some kind of remotely controllable robot to keep a moving 400 pound (in the US we have many obese patients) patient's airway open would not be easy.
Another other difficult problem would be nerve blocks: remotely advancing a needle to just outside the spinal cord, or just into the left brachial plexus seems like a difficult job to do by remote control.
So cool use of a remote connection, but I'm not worried that I'll be outsourced anytime soon. I will be physically present at 2am putting that epidural in your back for your labor pains.