Hand sanitiser doesn't usually cause soreness because the natural oils in a person's skin protect it from absorbing much of the alcohol content. However if you wash your hands with soap and water first, that removes much of the natural oils, leaving the skin more vulnerable to penetration by the alcohol (resulting in soreness).
Using hand sanitiser straight after using soap and water is madness, particularly when the skin involved is that of young children!
As Woofgang says, it should be 'one or the other', and most definitely not both!
Surgeons need exceptionally clean hands but, even though they go through a lengthy procedure and then don gloves, they still only use soap and water.
https://www.infectioncontroltoday.com/hand-hygiene/how-perform-surgical-hand-scrubs
'Aseptic' is always better than 'antiseptic'!