I wouldn't have, but if they've seen you straight away then there likely wasn't a huge queue of more urgent cases waiting.
Ask yourself though, what have they done that you could not do at home?
Yes its 3rd degree, white, no pain = third degree, you've killed the nerve endings. They may or may not regrow (probably not) but they're not vital in that particular patch of skin.
I'd have kept cooling it - then kept it covered and dry, changing dressings often to see if its blistering, opened up, or showing signs of infection.
If it did that or if it just opened up, I'd likely slap some honey on it and see where we go from there - if infection becomes more than redness around the immediate area, ie spreading outward or streaking, then medical attention as thats a spreading infection.
This is specific to that injury, in that location though - if it involved finger tips for example, or I couldn't move something that should move... then I'd have sought further help.
People should ask for help if its beyond their experience or out of their comfort zone, however 111 is often a sledgehammer to crack a nut and will over-react to tiny things simply because they can't see it, and don't know if you're underplaying it or not.