From what I understand, public money does not go to individual drug research, but goes towards more basic and foundational research. Drugs are built on top of the foundational knowledge, but the patents for those drugs do not then cover the foundational knowledge.

For example, if public research discovers what triggers cancer cells to multiply, drug companies will scramble to make drugs to prevent it. But the drug company cannot just patent the knowledge about the trigger. If a second drug company finds a second way to prevent the trigger, they can make it, no problem.

Maybe I'm just not aware of direct public funding to drug companies.

Because the profits from those drugs can be used to fund new drug therapies as well. Same reason as private companies. I'd prefer the drugs are made available to anyone who needs them at a reasonable rate (maybe a sliding scale based on income?) then making those discoveries public domain.

You need public money to help innovation but you still need individual's greed to make it happen. Would you spend your entire lifetime building something and you get nothing in return other than pat on your back ?