I don't think anything is solely the responsibility of the system. School should supplement what parents have already taught their child and vice versa, leaving everything up to school is not ok.
Just like Sex Ed it shouldn't be 'taught' it should be covered naturally as it comes up and in an age appropriate way. Have a one-off sit down talk is not good enough. No one remembers anything from that.
Early and Primary Years
Coin recognition
Piggy banks
Pocket money
Helping pay for things in shops with physical cash
Learning about 'Essential Bill' money and 'fun money' including being reassured that the essentials are always covered and that if a parent mentions not having enough money for X, Y, Z it's because sometimes you have to save up for the things you want and teaches delayed gratification. If you don't teach your child the difference then they go into school saying "mum says we haven't got any money" and the child brings home a food parcel that afternoon
Teenage years
Opening their own bank account
Get them to research various accounts available to them and the pros and cons of each one
Getting a job from 14
DD started working voluntarily at a charity shop 3 hours a week and I paid her £5 an hour. She has work skills ready for when she can move to proper paid work at 16 and has learnt financial skills.
Sharing your payslip
Show your child how much you earn and the deductions
Reviewing job advert
Look at salaries available for different roles Vs the job description
Put the salary into Salary Calculator to see what take home pay really looks like
Credit / Store cards
Teaching them about credit and store cards and the pros and cons of them
Mortgages
Looking at Rightmove at housecosts and how the deposit system works
Play around and see what salary would be required for which house
Financial decision making
Talk through your decision making processing on all big decisions including financial.
How you manage to balance those home repairs, big holiday, unexpected bill etc.
Various finance education tools are available online from different banks and other agencies