One of the controlling factors for socialized healthcare is that prices are negotiated down by the people paying for the medicine. In countries where private healthcare is extremely rare, pharmaceutical companies can choose between "less profit" or "no sales in that country at all". Sometimes they bluff and in rare cases that means public healthcare has to go without certain medication or certain vendors, but on the whole the price is kept under control (until corruption kicks in, at least).

When the people handing out cheques don't get a chance or don't bother to demand lower prices, things become incredibly expensive. Even if a party like a private insurer tries to negotiate the price down, the healthcare provider can always say "tough shit, guess your customers aren't insured then" as long as there's at least one insurance company willing to pay the full price.

You also see this with electric vehicle incentives. Governments incentivising people to buy electric cars by giving money directly to the consumer just end up with electric vehicles rising in cost because the money is essentially free anyway.