Actually I had an odd experience with this as well. My son is 15 now and went full cycle from 100% dependent to on a healthy allowance, but had to pay for a lot of his own stuff, even paying for his own ice cream, and back to basically 100% dependent. I honestly didn't see any value for the allowance. It started to look almost like welfare: any money he got, he used to spend on ice cream, video games, or things like that. I don't think that giving my son an allowance really taught him anything valuable about managing money, so we ended up discontinuing it.
Now he does still have some money of his own but basically he participates in budgeting. If he wants a new video game, we discuss what spending looks like, whether he has anything he can contribute towards the purchase, and basically explain it is part of the family entertainment budget. Money spent on ice cream or video games has much more value to him now that he sees that mom & dad have to give something up in order to spend the money on fun stuff, even if that thing they're giving up is intangible, such as putting money towards future savings.
He seems to understand the value of money better now, even though he's only indirectly spending it than he did when it was just showered upon him to spend however he saw fit. I'm guessing all kids are different, however. I think that's part of the challenge of parenting; you have to adjust the parenting style to what you see working with each child, not just a one-size fits all approach and you let them either pass or fail, similar to the school system. As a parent, you should never see failure as an option.
15 years old and you're still paying for his video games beyond "whether he has anything he can contribute towards the purchase? Out of the "family entertainment budget"? My son paid cash for a ~$500 acoustic guitar by that age--he saved cash from birthdays, Christmases, and any side gig he could find for a couple years to pull the money together. And that's to develop a talent that he's now used to play in churches, youth camps, school, and paid weddings (the weddings have reimbursed all of his musical expenditures thus far). $10-20 a pop for used games at GameStop or some other similar video game shop should not be difficult for your son to save to buy, and that experience will teach him far more than any "budgeting discussions"--namely, delayed gratification, saving towards a goal, work requires effort but leads to rewards, etc. Talking to either of my sons about how "I" budgeted to spend "my" money on "their" purchases did very little to help them--and they bored with that discussion quickly once they knew it wouldn't change the outcome in their favor. Their questions were simple and consistent--How much? Can I have it? When? They knew the 'how much.' I determined the 'can I have it' and they learned early on the answer was usually 'when YOU have the money.' The 'when' was 'when YOU have saved the money to buy it.'
Don't get me wrong. I provided the things in life my sons needed to be successful and have a great childhood. As a result, they have had experiences that none of their classmates have ever had, and they have demonstrated great work ethic from middle school on to college now. I just emphasized that outside a few special occasions, the fulfillment of their wants would be dependent on the money in their piggy bank, not the money in my bank account.