It’s not even just a “wildly unrealistic” promise

Dating back to Nixon's Economic Stabilization Act of 1970, Presidents do in fact have the authority to set price controls. And this was common practice through the Carter Administration. Under Reagan, the The Council on Wage and Price Stability (which was organized under a '74 amendment to the Econ Stabilization Act) got moved under the Office of Information and Regulatory Affairs. And its powers have largely gone unused by the generation of neoliberal Presidents and their economic advisors who succeeded it.

But it isn't wildly unrealistic nearly so much as it is wildly unpopular among DC pundits and ultra-orthodox free market economists.

it astonishes me that so many people believe he can drive down prices in light of the fact that he’s also proposing a thing whose specific purpose is to prop up prices

It's arguably the secret weapon he could use to rebut the threat of price-spikes under a tariff regime. Margins on retail goods are absurd - 60% to 80% of retail product prices find their way into some form of overhead or profit in the supply chain. An administration that did want to cap prices absolutely could drive down the real cost of consumer goods nationwide without actually hitting the labor/materials costs of those products.

However, they'd do this on threat of a capital strike. The handful of mega-corporate manufacturers, distributors, and retailers could respond to price ceilings by freezing activities and setting off a massive "Venezuela Style" supply shock that they'd then order their media wings to blame on the President's actions. This would foment a reactionary backlash and likely trigger waves of stochastic violence aimed at anyone who supported the price ceiling policy. At this point, the only play a president would have is mass nationalization and centralization of economic authority. And good luck with that.

That's the real Sword of Damocles hanging over the head of any executive daring enough to exert power against the corporate boards that ultimately govern the country.