This is far too simplistic and all but completely ignores the influence of deficit spending by government to compensate for sectoral balances and wealth accumulation. Too little deficit spending drives leveraging of private debt and fuels asset bubbles that eventually burst when the bills come due and “real” money is scarce. If banks loaned from reserves you might have more of a case for ignoring public sector spending in the mix, but they don’t. Concentrating on reducing public debt (that is totally misnamed and doesn’t require repayment) with austerity drives private debt with much higher interest and the requisite repayment.
Your description of inflation as an automatic response to money in circulation is completely off base with a fiat currency. The main indicator of inflationary pressure is always either 100% employment (real jobs, not multiple minimum wage jobs per person) or limited goods and services to absorb currency (WWII). We could double deficit spending for decades and likely not kick off significant inflation outside of a few bottlenecks in production. Japan has been trying to inflate its currency for about that long with no success in spite of a debt that is over 200% of GDP.