That is true all else remaining equal - but that isn't usually the case. Lower rates do increase the demand for governments generally to borrow as well at a faster rate via lower IR rates.

The amount of interest payments therefore long term (not short term) isn't really affected by the IR rate but more by politics and the amount of IR payments/debt burden they can politically get away with - in the US it is a LOT - in other countries the political appetite can be less.

So while it is true that higher IR payments do increase the money supply, generally with lower IRs governments are encouraged to "borrow more" by many stakeholders to their capacity under the low rate anyway. For example I saw many newspaper articles around our local media stating things like "rates are low, the government should invest that in infrastructure/disability programs/{insert favorite idea here}, etc when rates were low with politicians happy to spend accordingly.

In addition under low IR's the private sector will borrow more increasing the amount of credit in the economy as well - also inflationary money supply.

There's always nuances; these black and white theories can be dangerous. They assume all else is equal which is rarely ever is.