The author must live in a cheap part of the country or something (maybe the New England area where there are supposed to be rinks at every corner, haven't been there). For example for Prelim through Juv she says basically 7.5 hours of freestyles a week, and 2.5 hours of lessons a week (assuming she meant each private lesson is half an hour long). Where I'm at that's basically $112 of freestyles and $200 of coaching fees (depending on coach) per week, or about $16k a year right there. Then you got competitions, skates, tests, costumes, etc., yet she says it only comes out to $7k-10k a year.
OK so I was just doing the math for my skater and I think I am spending more also. I think you are all right about this article being written by someone who lives in an area where the costs for skating are low.
But remember too that you don't want to multiply by 52 weeks, even though that's how many weeks are in a year. Skaters are supposed to take a few weeks off here and there for vacation and rest. Skate Canada recommends that skaters train 44 weeks per year. Many other sports experts recommend similar things - they say the kids can do sports/activities all year long but should rotate and only do each sport about 3/4 of the year at most.
So here is my skater's math:
1-2 hours per day 4-5 days per week, freestyle sessions are $10 per hour (but there are discounts for purchasing blocks of 10 at a time and also skaters can get free freestyle passes by volunteering to help out with Learn To Skate) = $80 - $200 per week
3 lessons per week, each are $40 = $120/week
multiply by 44 weeks per year
= $8800 - $14080 per year not including any fees related to competition
We just went to a competition and hotel, gas, entry fees, and costume cost over $1000 for this one. That's not including coach's fees. Others we have been to cost less. The biggest factor for us is hotel costs and flights if necessary. (Right now, to save money and hassle, we are going to avoid competitions that require flights.) Anyway, 2-4 competitions per year for this level FOR US works out to anything from $1500 up to $3000. (We save money on costumes by getting used and DIY stuff. At this level it does not seem important to have fancy expensive costumes. And some of our competitions are local and thus no hotel or travel costs.)
Skates cost $800 this year. (But a friend has the same skates gently used and only paid $250 for them)
So at a minimum WE are spending about $10k per year at this level. Practically speaking though I think we're spending closer to $15k or more.
But if we really wanted to keep costs down, I do believe it would be possible to get the costs down to about $7k. The biggest savings I think would be by getting free freestyle passes through volunteering, which really can only be done by the older kids and adults.
( Edit to add: my area has coaches that charge anywhere from $22/30 min - $60/30 min. Generally, the rate is strongly correlated to the experience and quality of coaching. )