MMT is a way of looking at the financial part of the economy from a different point of view. It is often described as a Lens you can look through to see things that hither too have not been made clear.
When looking at the banking system, that view looks something like this
What this view highlights is the institutional and accounting realities of the financial system. It is hierarchical, and in the majority of the world the central bank is a creature of the sovereign legislature, who sets its remit - the playpen in which the central bank toddler can be ‘independent’.
Other entities operating in the currency zone form a hierarchy of balance sheets under the central bank. Only some have central bank accounts. Others do not. Those with central bank accounts must act as nominees for those without. This has an important effect. It means there is an extra set of financial liabilities and assets in the system for each layer in the hierarchy. If I have an account at a non-bank financial institution and I receive a government payment that will create an equivalent additional asset for the non-bank and the clearing bank.
For example: I get a 100GBP asset, the non-bank gets a 100GBP asset and the bank gets a 100GBP asset for the same single 100GBP transfer from HM Treasury. Everybody then gets to play asset swap games in the financial markets with their version of the asset. All three could bid at a Gilt auction for example for the single 100GBP Gilt on offer using what is essentially the same 100GBP.
That’s why the banking hierarchy matters and why it is vitally important to get the institutional structure and the accounting right. You cannot just consolidate the whole thing, abstract it away and then wonder why bond auctions keep getting covered bids.