I have no particular insight into the pricing of this drug, but in general, NIAID funding doesn't get anywhere close to the level of investment needed to bring a drug to market (NIAID's total budget is something like $5Bn. A single drug can easily require that much money to go to market).
Moreover, it isn't a bright-line distinction between public and private funding: if a novel drug is developed (entirely privately) against a target for which academic researchers once received public funds, does it count as public funding? There's a huge gray area in between. Nearly all drugs can be characterized as "publicly funded", if you look hard enough.
Rules like this sound great, but in practice, they'll lead to perverse incentives (like drug companies completely avoiding targets from the academic world, in the previous example). Nobody rational wants to spend billions of dollars doing late-stage development and clinical trials to be torpedoed by a researcher who took a small amount public funding decades prior.