Yes, it's limiting, but that's because anything outside of that definition of life would be purely speculative. On the other hand, we not only have an example model of how life can work, but reason to believe that other forms of life may be similar. Both carbon and oxygen have relatively unique traits that make advanced life possible without extreme inefficiency or spontaneous combustion.
Why isn't there life that relies on other gases like fluorine chlorine to perform oxidization? Besides the fact that oxygen is presently far more abundant than those other gases, oxygen (specifically dioxygen or O2) has very interesting and important characteristics; it has a highly energetic double-bond that's easier to break apart than other similar molecules, and it has both a stable state (triplet ground) and a highly reactive state (singlet). Without getting into the weeds, this has to do with the quantum property of the "spin" of its electrons. Life, including the human body, uses some neat tricks to be able to transport oxygen and switch it to be highly reactive where it's needed, particularly in the Krebs/TCA cycle. If oxygen didn't have both a stable and a reactive state (while being the same molecule!), it would either be a less effective oxidizer or be too reactive to be safely transported biologically, let alone to be exposed to. So there's reason to search for life in parts of the universe we know to have oxygen – it's really an amazing molecule.
Carbon might be more replaceable than oxygen (IMO), but even it so far seems to have the most advantages to life in contrast to other molecules, from what I understand. I'm sure someone will correct me on this (which I absolutely welcome!), but carbon is advantageous for life because it is almost an opposite of oxygen; it will accept multiple bonds with other molecules but doesn't need those bonds so badly that it needs to steal electrons from other molecules (oxidization) or go out of its way to lend its electrons, causing a redox reaction. It's a chill ass molecule that's really good for being a building block for far more complicated molecules.
Now let's talk about water. It's actually a really good solvent and can dissolve a wide range of substances, thus making it effective as a carrier of many things. Water is a polar molecule due to the way it has an oxygen end and a hydrogen end (think of the way an H20 molecule is normally represented as being triangular), which allows it to interact with other polar molecules (including other water molecules) and reject nonpolar molecules. At the same time, it's capable of interacting with some nonpolar molecules. By being polar, it forms hydrogen bonds with other molecules. These properties are important for things like the creation of proteins, for example. Water is also a side effect of organic chemistry; both respiration and the actual burning of organic material results in CO2 (and other carbon molecules) and water. Water stores heat very well, and temperature may be an inherently limiting factor to where organic life of any kind can thrive. Oh yeah, and it also has a neutral pH. Though it doesn't conduct electricity on its own, a solution of a water and an electrolyte allows it to conduct electricity. The ability to create balances of electrolytes is another critical use of water in organic life.
I'm missing a bunch of things, but the picture I'm trying to paint for you is that molecules such as water, carbon, and oxygen, aren't simply replaceable in the only example of life that we know of. Their role isn't by mere happenstance, but because they have inherent advantages over their alternatives.
Searching for life elsewhere in the universe is an expensive endeavor. While we shouldn't be close-minded to other models of life, searching for life similar to our own is simply more efficient from a logical perspective. It would be one thing if Earth was the only object circling a star that has these important molecules, but we know that it's not.
Something to consider is just how similar life is on our own planet. We like to think of some animals like octopuses as "alien", and while they are indeed strange in many ways, they still share properties with other animals. Take for instance the fact that octopuses have two eyes, just like other animals; in fact, the vast majority of animals have two eyes. Even spiders, which have more than two eyes, usually have eyes arranged in two rows, and many spiders have two eyes that are more dominant than the others. Eyes have evolved multiple times independently in the tree of life. While it's an interesting idea that extraterrestrials might have multiple eyes, there's reason to believe that, if we were to meet extraterrestrials that evolved on a similar planet to Earth, they would feature two eyes. Although we might not completely understand why evolutionary pressure hasn't widely promoted more than two eyes, it still appears to have an inherent advantage. This same line of thinking is why scientists have focused on searching for life similar to our own; there are properties to our form of life that are based on environments and traits that have inherent advantages. The evidence suggests that the way our life has arranged itself isn't merely a jumbling of various things that happen to function together.