Again we are thinking similarly, I've made that point myself, and again it says that too much is made of the "measurement problem" in an expressly quantum mechanical situation (as a similar problem exists classically).Consider, for example classical mechanics. Imagine, you have a ideal gas, consisting of tiny balls in the box, which behave according to newton's laws. If you want to do any measurement of the state of the gas inside, you also have to interact with it somehow, introducing some contact with outer world, which is in thermal state, thus affecting the gas in random manner. Thus,the situation is IMHO pretty much the same.
I agree that we are led to the same conclusion both ways, but where we disagree may be our conclusion about reality. We both have used the word "deterministic" to describe a theory, not necessarily a reality, because determinism is itself an aspect of a model. As with any model, the task of comparing it to reality falls to us, and we determine how we will make that connection. The concept of "determinism" never survives that connection, it is lost in how we do science.However, in this case, I personally would consider such a system deterministic. The quantum mechanics is for me just different, more sophisticated way to establish the equations of motion.
So neither theory "tells us" that reality is deterministic, though both theories are themselves deterministic until they make a testable prediction. The testable prediction, by virtue of the testability, requires that contact is made with elements that are outside the theory, and those elements will introduce the concept of randomness. We agree there. Where we may disagree is that you seem to view that as a kind of side effect of testing theories (as when you said "everything is deterministic", it's not clear what you meant there), but since the whole point of a theory is to be tested, I do not distinguish the expectation that the theory must be testable from the theory itself. In that sense, no theory is truly deterministic once its encounter with reality is included in the grand picture-- it is only the theory as idealization (i.e., as a mathematically structured object) that is deterministic.
Hence, everything is not deterministic, but key elements of the models are. We recognize that the wave function will evolve deterministically, but when we go to test it, we will encounter an incomplete ability to predict the testable outcome, and that incompleteness will obey an uncertainty relation (as alluded to by peter0302 as well). Some would count that as a lack of determinism, not so much in the theory itself, but in its point of contact with the rest of the scientific exercise. Others see that apparent breakdown in determinism as so severe that they feel the need to outfit an exostructure of "many worlds" just to avoid it, but you and I can question as to whether or not that is really necessary.