I really don't blame scientists for this, as you say it is their job to discover new things, and I think many do that knowing that regulating the safety of their discoveries (or regulating what they should be allowed to research) is someone else's job. Someone really should be doing that job - but clearly they're not.
It is corporations that essentially pull the trigger and do the damage through the widespread adoption of potentially unsafe technology in order to gather more profit. I think most people have accepted that corporations are fundamentally evil structures, and so it's no surprise at all that they'll murder babies if you let them.
I see the primary failure one of regulation, which is the government's job. Unfortunately the voting public has allowed government to be captured by psychopathic corporations. I don't think there's a simple fix for this, or any isolated weak point to blame. From a distance it's just what we are as a species.
You're right that it is not the scientific method, but rather the institutions and culture of scientific research. The scientific method doesn't necessarily lead you to any specific sorts of behaviors like this, but the structure, culture, and incentives of scientific institutions sure do. The precautionary principle is abandoned when scientists push forward without adequately addressing possible harms. Usually the adoption of new technologies goes something like theoretical science -> applied science -> engineering -> business. So for plastics, for example, you didn't have some CEO one morning wake up and just say, "I've discovered this new material we will make everything out of, risks be damned." The failure to investigate harms and safety must have happened somewhere upstream.
Instead of discovering mysterious substance A and not moving forward until investigating safety, usually this works by the scientists focusing instead on discovering substances B which is 5% better (cheaper, stronger, etc) than substance A, then focusing on substances C, D, etc to keep improving it. The engineers hear about substance B which isn't quite practical for use, but it is close, so when substance E comes along they add it into their new product, and the business is happy to have this new product and capabilities. What a competitive differentiator this will be in the market for us! The safety failure almost always happens upstream of business--in a charitable scenario they just trust the institutions of engineering and science upstream of this will build something safe.
Economic and business incentives certainly have a role in this, but scientists willingly go along with all of it because it suits their needs, research, and objectives. I don't see much better ethics in scientific fields than business, to be honest; it's still humans either way. There are plenty of conscientious, cautious, and reflective people in both disciplines and also plenty of predatory sociopaths. Scientists will gladly publish all kinds of things of questionable ethics and societal implications. The most important thing for them is making the new discovery and publishing. It's invention for invention's sake. Discover something new and deal with the harms later. In fact, those will make some great papers building on this! Lots of times those things turn out to be interesting and useful, so any harms created by the process are more or less dismissed as acceptable costs of doing business.
People "trust the science" and government that it is safe. There is nonstop propaganda from every institution of power (universities, government, businesses, law enforcement, military, media, etc) echoing this message that those with power are competent, trustworthy, etc. Sure, we might expose some problems here and there, but there are always some bad apples everywhere, and isn't it great that we have such great enforcement institutions to catch them? All these institutions have incentives to parrot the messaging from other institutions of power since that shared image of legitimacy boosts themselves. In a way, these institutions operate as a collaborate cartel. So I understand why people put that faith in these institutions. But fundamentally trust is an unscientific concept, so that was the mistake.