I do not think it is that difficult, if we separate in two cases:
Outside of a scentific approach: we are dealing in that case with opinions, speculation, with faith, etc. There is nothing we can say scientifically about the fact. A priori, it does not imply anything "good" or "bad" about the statement. As you say, it may or may not exist.
Within a scientific approach: it is rather frequent that a scientific theory that has been demonstrated by experiment to be highly workable is predicting phenomena that experience has not measured yet. This has happened many times in the past (in chemistry, in physics, etc.: e.g. some new elements or "black holes") that those predictions have been confirmed. Apparently, quantum mechanics is full of those enlightened guesses which are surprisingly helpful in doing predictions. Sometimes they haven't, of course, and experiment turned out something different.
Perception and Prediction
Actually, prediction is at the core of the scientific method, because it is a very important criterion in determining whether a theory is valid or not. If researchers, scientists or engineers (when working on the drawing board) could not indulge in the highly productive activity of prediction because 'it does not exist', then this would slow down the progress of science and techniques. In particular, it would oblige researchers to do very costly experiments upfront to measure everything, instead of calculations (simulations) that could give an indication whether it is worth continuing in a stated direction. And apparently, quantum mechanics is full of enlightened guesses which are surprisingly helpful to make calculations and predictions.
More broadly, if we considered that what is not observable does not exist by the mere fact it is not observable, we would live in a very flimsy reality indeed. First of all, what do we mean exactly by observable? In astronomy, many planets that are considered to exist have never been observed but have been "calculated into existence" because they seem the best explanation in the variations in observational data. The same applies to black holes, which are not observable by definition because they emit no light; what is observable is indirect effects.
Difficulties of prediction
The traditional mecanical world (Newtonian) could perhaps have open and shut discussions about this ("I believe only what I see or touch"); but with the onslaught of new phenomena which are far too big for our senses to perceive or sometimes even to grasp for our mind, or conversely too small, the notion of what we call reality had to be relativized.
In fact, science did not even need quantum mechanics to get to that conclusion:
It is usually thought that this indeterminacy, that we cannot predict the future, is a quantum-mechanical thing, and this is said to explain the behaviour of the mind, feelings of free will, etc. But if the world were classical -- if the laws of mechanics were classical -- it is not quite obvious that the mind would not feel more or less the same. (...) Speaking more precisely, given an arbitrary accuracy, no matter how precise, one can find a time long enough that we cannot make predictions valid for that long a time." - Richard Feynman
Science versus philosophy
In any case, by pursuing existence in science, we could get into particularly convoluted paradoxes: for most people, the workings of computers, astronomy, quantum mechanics, even a car engine are a matter of faith. They have never looked into a CPU or a piston, they have only been told about it. Does it exist for them, since they never perceived it? It depends on the definitions.
Even in the daily life of this automated workd, quite a lot of things that we assume (or "predict") are there, we never get to experience directly. But we do assume they are "real" because it fits, it works and we are so rarely disappointed about them that we hardly give them a second thought. When book an airline ticket via the Web, we take an enormous amount of things on faith (a money transaction, an airline, an airplane, a seat, a crew, an airport etc.) and the only thing we have is a confirmation. How do we know, at that time, that they really exist?
To avoid such quandaries, one solution for scientists is to stay away from philosophical questions of ontology: to content oneself with (direct or indirect) perception and prediction, and to admit that (scientific) reality is some kind of enlightened social consensus about how things (are supposed to) work.