I've never heard that term before and I have skied at different slopes on the east coast. (NY/NJ/VT/ME/VA/NC) I see you're from NC; I've been to Sugar Mountain and Sapphire Valley. While we were asked if it was our first time skiing, no one suggested "Never-Ever" lessons.
In skating, we call newbies "beginners." It's perfectly good word, but I'm sure some people would get their back up about being termed a "beginner" at anything...it could be a regional term or one that the ski resort/industry copyrighted, but to me, it wasn't obvious what you were asking. Adding the hyphen in your post made it more clear than the subject line. YMMV
For most people, skiing involves an intentional road trip to the mountains, usually accompanied by a hotel/resort stay unless you live within a reasonable driving distance from a resort. I only ski once or twice a year now, but we used to ski regularly at Vernon Valley, NJ because it was about an hour's trip, so it didn't involve overnight stays. Some of my friends had season lift tickets and would go 2-3 times a week while their kids were in school If you've made the effort to travel overnight, it makes sense to take a lesson if you've never skied before.
However, skating rinks are rarely "destinations" - most ice tourists are locals who can always say "tried it - let's go out for ice cream." Many people skated as kids so they don't realize that it can be difficult. They fancy themselves capable of helping/teaching their kids, or they're from the "throw the kid in the lake and they'll learn to swim on their own" school. So they come and bring the kids.
Having done both sports in different places, I don't think skating is as dangerous as skiing, imo. A beginner can hold onto the rink wall and get around, whereas they have to ski downhill to leave the ski slope. If they struggle with skating or become afraid, they just have to go around the rink to get to the door; most of the hot doggers avoid the walls for that reason. On a ski slope, you have to sit-ski or walk all the way down the hill and hope no one crashes into you and that you can stop at the bottom. Go too far off the slope and you're in the trees or the lift lane, which adds obstacles to the fun. (I'm only half-kidding. It's amazing how many people think that the trail ratings are suggestions and end up sliding down hills on their butts.) Plus, conditions are far more controlled at a rink vs. outside on a mountain. That's why I prefer skiing in Vermont to NC; man-made snow is just scary because it packs so quickly and becomes icy as soon as the sun starts to get low in the sky. Add in a sudden bit of rain and you have to snowplow all the way down the hill or you will be skiing at breakneck speed, literally!
Additionally, the cost of skiing is so much higher that people feel committed to getting the most out of a trip to the slopes. Skating costs around $20pp with rentals and snack bar; it's not as painful to walk away if you're not having a good time. Many people don't feel the need to add onto the expense; there was a birthday party at the rink yesterday and an adult guest was struggling to skate and hold up a toddler at the same time. I suggested a skate frame and she said "No, they cost extra." I thought, "Well, it's worth $5-10 to save your back." but I kept my mouth shut and just sympathized.
I kind of like the walk-on lesson idea; it's the logistics that make it tricky at our rink because the front desk collects all lesson fees. I'd have to send someone off to the desk to pay before the lesson. It's just awkward, not impossible.