You'd think the government would be happy to save money...

Back on 10th March I bit the bullet I went online and signed on for Job Seekers Allowance (should have done it the day I was made redundant, but hey ho). 23rd March I had an appointment at the local job centre to confirm things and was then told I had to go back on 9th April, ie two weeks later, to present myself and confirm that I was still unemployed - apparently one needs to present oneself at the Job Centre every 14 days.
Anyway, this morning I called the National Job Centre number to let them know that I have now signed a contract and that I'll be starting employment again on the 14th, so please can they stop my claims etc. Especially important as, for the new job, I had ticked a box on the HMRC New Starter document stating that I have not received any money by way of income, JSA or Universal Credit (or similar) since 1st April (because I haven't received anything yet).
Anyway turns out that apparently the first payment to me has been processed and I should receive a letter confirming within the next day or so; but what I find totally bizarre is that in order for me to cancel / stop my JSA, I still have to attend the appointment on the 9th, tell them I have a job so they can deactivate me on the local system and then I need to call the National Job Centre phone number again so they can deactivate me on the Government system; and until that is done they cannot stop payments. It's just madness - had they been able to just press a button today and 'close my case' the government could have saved about £120 or so. Admittedly it's not the hugest amount of money, but I cannot be the only person in this country in the same situation; so multiply that £120 by who knows how many people, surely that would be such an easy way to save some money and put that towards the NHS?
So now, I for once get something from the government, and rather embarrassingly have had to email new job and, with huge apologies, ask them to please change the tick box their end
Anyway, this morning I called the National Job Centre number to let them know that I have now signed a contract and that I'll be starting employment again on the 14th, so please can they stop my claims etc. Especially important as, for the new job, I had ticked a box on the HMRC New Starter document stating that I have not received any money by way of income, JSA or Universal Credit (or similar) since 1st April (because I haven't received anything yet).
Anyway turns out that apparently the first payment to me has been processed and I should receive a letter confirming within the next day or so; but what I find totally bizarre is that in order for me to cancel / stop my JSA, I still have to attend the appointment on the 9th, tell them I have a job so they can deactivate me on the local system and then I need to call the National Job Centre phone number again so they can deactivate me on the Government system; and until that is done they cannot stop payments. It's just madness - had they been able to just press a button today and 'close my case' the government could have saved about £120 or so. Admittedly it's not the hugest amount of money, but I cannot be the only person in this country in the same situation; so multiply that £120 by who knows how many people, surely that would be such an easy way to save some money and put that towards the NHS?
So now, I for once get something from the government, and rather embarrassingly have had to email new job and, with huge apologies, ask them to please change the tick box their end