No, trickle-down refers very specifically to tax breaks tailored narrowly to the wealthy, in hopes that they will in turn distribute their wealth via increased employment of capital and, ultimately, labor. Since its origins in Gilded Age populist political writing and depression era popular media, it has never meant anything else, and it has nothing to do with the subject at hand.

> Give companies, who are additionally acting unethically, a guaranteed and direct profit increase...

No, lowering barriers to competition doesn't give anyone guaranteed profit. It reliably does the opposite, depriving incumbent companies of their monopolies.

> Furthermore, private corps aren't the only way drugs can be invented.

Nobody -- literally nobody -- would dispute that. The question is whether the government is able to do the various parts of drug development as efficiently as the private sector. This is not a simple analysis, but I think the answer is more often than not, No.