Ok. The relative risk depends on the underlying health of the child and the age.
What I tell all parents of my paediatric patients is that anaesthetics are inherently safe for children who are otherwise fit and well (normal pregnancy, not too prem at birth, appropriate milestones for age, no serious heart/kidney/lung disease [mild asthma is fine], non smoking parents). For these children, the risks they can face are : sore throat on waking, post op sickness, dental damage (if getting into the mouth is tricky - it isn't normally) and rarely, a reaction to the meds (1:10000 - 1:20000).
For an EUA, the chance of needing the medicine with the allergy risk is very low. The anaesthetic will be very short because the procedure normally is. The IV can be inserted asleep (if one is needed - we don't always use them, again depending on age of child. I do with nearly all cases.)
Depending on the age of the child and how well they go off to sleep, they wake up grumpy and cross - normally resolved with a drink.
Kids with underlying medical problems obviously are slightly more tricky depending on their medical history. Those who have parents that smoke are more likely to cough on waking (in the same way that smokers are more likely to cough on waking).
Anaesthetics are not "lighter" than they used to be. The drugs have evolved over the years to be shorter acting and the monitoring is far better. They are certainly nothing like sedation (which requires verbal contact to be maintained throughout the procedure).
You're not unreasonable to worry - I would if it were my child! I am sure it will all be ok though.