Councils do keep 100% of the council tax they receive, Khandro. You may be thinking of Business Rates (which are remitted in total to central government).
Just a glance at my local council’s balance sheet for last year shows income from council tax is £132m, central government funding is £425m. This means just 23% of their revenue comes from council tax and there is very little point in them sending the £132m to central government only to have it returned as part of the ££557m total. I imagine most other local authorities will show similar figures.
If you look about halfway down this document:
https://www.gov.uk/government/policies/giving-local-authorities-more-control-over-how-they-spend-public-money-in-their-area--2
You will see:
Local authorities fund their activities from 3 main sources:
grants from central government
Council Tax
other locally generated fees and charges for services
No mention that they hand over their council tax receipts to have them handed back either in full or in part. There’s plenty of places you can have this confirmed.