You're saying the cause is an increase in household debt? That would seem to cause inflation, not keep it low?
Suppose you have an existing $500K variable rate mortgage. Interest rates increase by 0.5%, you have to transfer $2500/year from buying stuff to paying interest, reducing inflation. If you had had a $200K mortgage it would only have been $1000/year for the same rate increase.
The high level of existing debt has made the rate increases unusually powerful at fighting inflation.
Then there's another question, if someone is paying more, it means someone else is getting paid more. So, while some have less disposable income, others have more. The overall net effect on spending can still be negative, but it's not an automatic conclusion that one can simply jump to.
But the someone else is the bank. From there it typically goes into government bonds or some such thing with an equally tenuous connection to consumer prices. Banks don't have a large desire for fresh fish or motor fuel that goes unmet only for a previous lack of funds.
What they'd like to do is find someone new to loan the money to, but higher interest rates make people want to borrow less money, so the bank will often just use the money to reduce the amount it itself has to borrow from the Fed.