AIBU to start regular pocket money for 9 to 10-year-olds?

Mine get £3 a week. They get to choose what they spend it on. It's great actually as they often decide not to buy the thing as it's out of their own money. I do limit sweets to one purchase per day. I've talked about saving, the eldest is better at it. Totally separate from chores as they should be helping around the house to be helpful not for the cash. The youngest is more of a spender but I think she'll learn with time. I'm considering giving them each another £2 a week and specifying that £1 is for saving in a savings account and £1 for charity of their choice. Eldest has just opened a bank account which they'll operate themselves.

My 8 year old has been receiving £5 a week for the last year. He has a monzo, he has his own card so pays for some of his own things and is interested in how much he has.

We have a big holiday coming up later in the year and we have decided to split the money into pocket money and travel money savings. So we pop £2.50 a week into a savings account. This has opened up a conversation about savings and interest etc.

It has helped him take ownership and understand how much things costs and the consequences of buying something more expensive etc.

This works well for us.

My daughter got rooster cards for her children and I transfer pocket money to her weekly which she puts on the cards. They get £1 for each year of age. She can monitor their spending and they are learning that they can save up for things they want. The eleven year old now goes on very occasional (supervised) shopping trips with her friends. She is learning to consider her purchases very carefully because it's her money buying them. The little one still needs guidance as she's never met a piece of plastic tat she didn't covet but she's become really interested in counting the cost of these items, if she has enough on her card and calculating what will be left on her card if she buys them.

This is something we've worked out through a fair bit of trial and error with my two boys. We settled on a system where the everyday household jobs, tidying their room, helping clear the table, that kind of thing, are just part of being in the family and aren't paid. But we've also got separate earning jobs, bigger tasks they can choose to do for extra money on top of their regular pocket money. That split has worked well for us because pocket money isn't purely a reward for chores, but there's still a way for them to earn more if they're saving up for something specific.
For the money itself we use four jars, spend, save, give and invest. One of my sons is naturally really good at saving and investing, though even he loosens up a bit around holiday time and puts more into his spend jar than usual, which I think is fair enough. What I really like is he keeps track of how much work he's put in that month against how much pocket money he's actually earned, and if he gets to the end of the month and hasn't got much, he'll come and ask if there are any jobs going so he can top it up. I always go along with it, because it's exactly the mindset I want him to have, that if he wants more money he knows he has to go and do something to get it, rather than just expecting it to appear.
Your own story about the school lunch money and the leftover change is such a good example of exactly why this stuff sticks best when it's their own money and their own choices, not something you're managing on their behalf. Sounds like you're definitely onto something wanting to hand over a bit more ownership at this age.
I'm a primary teacher and dad of two boys, and this is honestly the sort of thing I spend a lot of my time thinking about. I've actually put together a free pocket money guide that goes through a system like this in a bit more detail if it would be useful, happy to share it if you fancy it. I've looked at the research for average pocket money per age in UK, and explain the earning jobs and jar system in more detail in the guide :)

Mine get 20x their age per week (grand total of £2 for a 10yo) that I have an auto transfer into a separate account of mine.

It's mostly used for all the things I wouldn't buy on holiday like mr whippy and tat from the gift shop. But sometimes one of them will see something in the supermarket or want nicer clothes than I would usually buy. My 12yo put £10 towards a nicer school bag.

Once my 12yo turned 12 she got a bank card in her name that I control and I transfer money from her account onto that so she can buy sweets etc from the shop on the way home or on a school trip or pay for the bus. She's a saver so it doesn't happen much.

My 9yo likes to buy things online like Lego kits and craft things. She's less discerning but improving.

If they get cash for their birthday they usually like to keep it as cash but sometimes I'll take it and put the equivalent in 'their' account.

We start pocket money around age eight/Year 4 and they get half their age per week, so for an eight year old they get £4.

However, only half of this (£2) is for them to spend. 10% (40p) goes to a charity of their choice and the rest (£1.60) goes into long term savings that they are not allowed to touch. We give everything in cash so that they can "see" it (this might not work if the child would be overwhelmed with temptation!) and it's not connected to chores, chores are just expected generally because they live here.