This is actually a well published mis-truth for a couple reasons - more slight of hand. Anyone that says the fed can't raise rates due to the public debt doesn't understand central banking.
FIrst, the national debt is denominated in treasuries anywhere from 30 days to 30 years. Last I looked average issuance was 7 years, so even if rates rise we won't see the affect on the budget for years to come.
Second, the government doesn't make monthly interest payments like the rest of us, they sell bonds which need to be paid at some future date, with an applied interest rate. For example if the government sells a bond worth $1M a year from now today for $900K, in one year they will have to repay the $1M - at which time the realized interest rate was approximately 10% (not quite, but close enough).
Lastly the government really never pays any debt - it simply rolls the maturing debt into new debt / bonds - so the interest rates is compounded on a balance sheet, but its really just a number in the ledger.
The biggest near term affect of rising interest rate is lowered borrowing by the private sector. The government likely doesn't consider it at all when deciding how much to spend.