That's even more crucial when you're trying to vaccinate literally billions of people. Even for extremely safe vaccines (like these are), very low percentage bad outcomes can mean a lot of people willing to sue.
Billions of people also means a lot of cranks willing to call conspiracy theories in a lot of jurisdictions.
But leaving aside pragmatics, you're assuming that the manufacturers wouldn't be liable. If they'd e.g. hid bad results from some study, they'd presumably still be criminally liable.
Like, pretty much everything has some side-effects, and if (for example) you have the choice between doing nothing and having a million people die and doing something for which everyone is certain that that will save that million people but on average might kill ten other people, then asserting liability effectively means saying that you shouldn't do that thing, because you won't be liable for the former million deaths but will be liable for the latter ten ones. So if the society wants people to choose the former option, they shouldn't be liable for such consequences.
That’s exactly why the idea was to get the government to pay for it and then force everyone to take it.
But it's a big leap to go from "Pharma is driven by profits, therefore they should be put under more scrutiny" and "Pharma is driven by profits, therefore they are injecting us with 5G-enabled nanobots to enable mind control by the Council of Rome"
They are put under way more scrutiny than other industries. Or where is the FDA equivalent demanding phase I, II, and III trials for so-called self-driving? Or imagine a pharma company embracing the slogan "Move fast and break things."
If you want an evil corporation, it's the small ones that are usually the worst.