If you want children to understand how to budget and save, they need an allowance.
I presume that parents who don't give their children an allowance pay for stuff themselves. For example, kids need clothes--if they don't have an allowance the parents are buying the clothes. Or if the kid wants to go to a movie with some friends the parent pays. If the kid gets a toy the parent pays.
If you pay for everything your kid gets yourself, whether you are frugal and buy them very little or are spendy and buy them a lot, they will not internalize how to handle money. For example, let's say the kid wants an xbox. There are two scenarios--the kid who gets an allowance and the kid who doesn't.
Scenario 1: The kid asks for an xbox, parent says "you can have one if you buy it yourself." Either the kid has enough money to pay for the xbox or they don't and they have to save. Then the kid has to decide whether they want to spend their saved up money on the xbox or not. If they do spend it, they have less money for the future. So they learn about delayed gratification (from saving up) and basic budgeting (if you spend money on one thing you have less money for other stuff). If they decide not to get an xbox there is a reward--they get to keep all that saved up money. Then the kid has learned about frugality and saving. Either way, the whole process has taught the kid about being responsible and managing their money.
Scenario 2: The kid asks for an xbox. The parent either buys the kid the xbox or doesn't. If the parent refuses to buy it, the kid hasn't learned anything about money--as far as they're concerned they didn't get the xbox because their parent is mean, not because the kid spent their money on other stuff. If the parent refuses to buy the xbox the kid doesn't get to keep the money, so there's no incentive for the kid to deny themself and be frugal. The kid learns nothing managing money. Or, the parent does buy the kid the xbox, and the kid learns that the best way to get stuff is to convince their parents (instead of saving up their own money). In addition, the kid doesn't have to sacrifice any of their own money for the xbox, so they learn that stuff is free for them. Either way, the focus is on whether the kid can convince their parent to buy something, not on whether the kid can save up money to buy something themself.
My parents gave me an allowance from when I was about 5 years old. It started out just a quarter, which I used to buy candy (my parents never ever bought me candy). Once it was a bit higher (a few dollars) my parents required me to divide it into three categories--long-term savings, charity, and spending. The long-term savings went into a bank account that I didn't touch (I also put gifts from relatives into this account, along with earnings from lemonade stands,and when I was older, from my part-time job). The charity money went into an envelope until it was enough to donate (I got to pick the charity). The "spending" money was mine to do whatever I wanted with. I could spend it right away, or save it for later. I mostly used it to buy books (in addition to the many library books I checked out every week). If I wanted to buy something, rather than my parents buying it for me or refusing outright, they would just tell me to buy it myself if I wanted. This taught me to take responsibility and really consider whether I wanted something enough to buy it.
That bank account was enough to buy a (cheap) car when I was sixteen. I used up all the money but since I had a job by then I quickly saved up a lot more money, which I could put in a Roth IRA. All those years of saving made it easy for me to just stick my paycheck into a savings account (and then transfer to the Roth) instead of blowing it on unnecessary stuff. If my parents had always bought stuff, or refused to buy stuff, for me, I probably would have spent all my earnings on clothes or starbucks or whatever, since I would never have learned that savings paid off.
As for chores, my parents didn't tie the base allowance to completing chores because they wanted to teach us that doing chores was a necessary part of contributing to the household, not a way to earn extra cash (so there was no option to refuse to do chores and forgo the allowance). We could do extra chores for more money, but we had to do our regular assigned chores no matter what.