Science is intended more to adapt an actual "theory" over time to better suit the evidence that it is presented with until it increasingly encompasses all edge cases that relate to the topic in question. That "adaption" can be considered disproving with an immediate re-creation of an alternate theory moments later to encompass the changing circumstances. In that narrow world view, than yes, disproving of a scientific theory can happen quite regularly, simply because there's a LOT of science going on.
On the flip side, actually "proving" something is exceptionally hard work. It is saying that at no point, ever, under any circumstances in this or any conceivable universe, with any natural or unnatural influence could this situation *EVER* take place for *ANY* reason. These are the rules, these are how things behave, and this is how things will always, and forever behave; EXACTLY like this and there's not a damn thing that anyone including the hand of God himself could do to change that.
Now think about that for a second and the level of difficulty involved in actually "proving" something and considering it "proved", solved forever and ever, and letting us as a species move on to bigger and better things. And that's ultimately the problem. Saying that something is "proved" means that there is nothing more that could ever be known about that topic, and that nothing could ever impact that field, be it further advances anywhere else, supernatural influence, extra dimensional characteristics, weird things that we haven't even considered possible... In most cases a theory remains "good enough". Gravity is one such theory. We know that it exists, we know how it works, we know how to calculate it, we know how to utilize it's traits for all kinds of things. But "proving" that water goes downhill ... It's something that we take for granted and require to base civilization as a whole on, through irrigation and plumbing. Something doesn't need to be "proved" to be immeasurably useful in the daily lives of incalculable people over countless generations. You may think that this is getting pedantic, and it is, but at the same point, it is the difference in Science between "Proving" a theory and not.
Referring to a simple and previously untested idea as what you've described in your swan scenario as a "theory" is what is ultimately damaging the credibility of the term in public perception.