Strong/weak empiricism is a false distinction in an important sense: the latter is the only option. Strong empiricism is dogmatic because it asserts more than can be demonstrated - it is upheld irrespective of evidence. As you know, this position is self-contradictory in that it says everything we know is experienced, and itself professes to be known, but it cannot be demonstrated with empirical evidence. Either we accept that dogmatic empiricism is held in faith and may be true, or we modify it so that it does not ask for more than empirical evidence can provide.
Why can’t we prove this empiricism certainly? We do have evidence for it, but only inductive evidence, which you also know is not enough for certainty. Therefore, the only possible empiricism is one which does not ask for anything more than reason to believe, because empirical evidence is capable of providing this, therefore ensuring that the proposition is not held irrespective of evidence (dogmatically). You say that this cannot be done because empiricism cannot be formulated as an empirical hypothesis. ‘Empirical’ isn’t entailed by ‘hypothesis’, and even saying it is a hypothesis achieves nothing, in your view. Firstly, what is meant by hypothesis is that the proposition is dependent upon evidence and subject to revision by it; so it is not the content of the statement but the way it is treated. We have to approach it with empirical evidence, else we fail to recognise it as a hypothesis, and it is only evidence that permits us to propose it in the first place. Empiricism is this: It seems (empirically) that there is at least one necessary condition for knowledge, and this is reliance upon experience.
Empiricism does not say “Most often experience is required for knowledge, but not always” because this is not what evidence suggests. Just like evidence does not suggest that “gravity nearly always draws mass together”. This is because all the evidence is in favour of empiricism. From that we can say provisionally that all knowledge requires experience. That is the theory or hypothesis. As you say, scientists observe all kinds of examples of phenomena, but they do not conclude finally and universally based on these observations. This is purely the problem of induction, and it means that what we hold as certainly true beliefs we should actually refer to as ‘propositions we have reason to believe in’. If you think that is not good enough and actually most knowledge is held more certainly, that is another discussion, but I disagree.
Your second problem is also interesting, but it doesn’t impact on empiricism. ‘You are using reason to assume that objects in thought are objects in reality’. However, empiricism relies solely on perception to tell us about perception. There is no necessary distinction between perception and reality; we only have our perception to access the world, our perception to live by, and our perception to inform us of the success of any hypothesis. Reason and empiricism are not separate either, many empiricists think that rationalist thought tells us a lot about how we come to think things, but does not provide us with knowledge of the world. Reasoning tells us that it is logically impossible to hold that perception is illusory, because we arrive at that conclusion from the evidence of experience. We cannot use one perception to undermine perception: self-contradictory.
From this, I hope you can see that empiricism is plausible, depending on evidence, and that it is not ‘weak’ simply because its more ambitious brother is a logical failure.
