Why not just print?  They need to keep the system credible (as much as possible, anyway) so the elites can continue to benefit from it.

Borrowing, rather than printing, gives the impression that the government will shape up and pay back.  The fact that, in practice, the repayment will never really happen will only slowly dawn on people.  By that time the original leaders have already made out nicely.

Also, by keeping public debt and money separate, the elites keep open the option that they will support faith in money but abandon public debt (and in fact supporting money *by* abandoning public debt,) in a crisis.  They are hoping that the threat of this will keep the government less wasteful, and thus prolong the system's lifespan.

That said, the Fed does print a little from time to time and give it to the government.  Quantitative easing is one example, loosening monetary policy with "open market operations" (printing money to buy government debt to influence interest rates) is another.  But the Fed remains a minority holder of Treasuries, for the above reasons, at least for the time being.