Bentham's utilitarianism, was centered in the "greatest happiness to the greatest number". He refused to consider any criteria to qualify the action in itself, because the validity of that action was depending on its results.
If the result of an action brings greater happiness for the large number, the action is good.
I am far from an expert on Bentham, but I feel you have taken "the greatest happiness to the greatest number" out of context, and misunderstood it a little.
The greatest happiness, surely, is complete happiness, and the greatest number is all. So saying we should strive for "the greatest happiness for the greatest number" means we should strive for "complete happiness for all". Not especially controversial these days, but don't forget that when Bentham was writing in Britain it would have been more popular to say "happiness, to the point where it remains decent, for myself, my family, and perhaps others of my class".
So, "the greatest happiness for the greatest number" is an fairly uncontroversial aspiration. Where it starts to get interesting, difficult, and controversial, where you have to decide how to balance priorities. Which is more important? The greatest happiness or
the greatest number?
As you say, utilitarianism refuses to consider any criteria to qualify the action in itself, because the validity of that action depends on its results. Do you disagree with this? Herein, I think lies the first discussion point.
That leads us to serious problems. If slavery brings greater happiness for 90% of citizens of a country, does that mean that is good to enslave the other 10%?
Because of the Holocaust, Jews received a state, a nation. That makes Holocaust good?
These criticisms do not seem to me to follow from "the greatest happiness of the greatest number". Clearly, 90% is not the greatest number, it's a huge 10% short! It assumes a method of balancing interests which we haven't yet discussed. Although I don't think Mill ever explicitly states it, it seems heavily implied in his writings that the first aim of utilitarianism is to minimise pain. Pleasure and pain cannot be weighed against one another, any pain is only acceptable if all other options would result in more pain.
In you slavery example, 10% of the population being enslaved would only be an OK result for many (perhaps most) utilitarians if all other possible options would result in more than 10% of the population being enslaved, or 10% enslaved for longer.
Your second example seems to me a critic of another soundbite version of utilitarianism; "the ends justifies the means". I find this statement misleading, because it implies that some events are 'means' and others are 'ends', whereas in fact all are both. So the Jews receiving a state is an ends, and a means, as it the holocaust. Many millions of Jews suffering and dying is, surely, a very very bad ends. It doesn't stop being a bad ends (or, rather, consequence) just because there are other, later, ends that might be good. So:
Q: Because of the Holocaust, Jews received a state, a nation. That makes Holocaust good?
A: No. The holocaust is still the terrible consequence. The Jews receiving a state is (perhaps) a good consequence, but does not weigh against the terribleness of the holocaust.
The holocaust could only ever possible be called 'good' (or, rather, 'more good' since only complete happiness for all would be 'good') if all other possible outcomes were worse (more millions being killed, for example).
The problem with Bentham is that it is not easy to follow the consequences of an action. Some of those consequences are in a logical sequency of the action, but others are reactions to that action - as in the case of the creation of the state of Israel.
It certainly isn't easy to follow the consequences. But this is true of many, many areas of science. For a doctor, for example, the aim of a particular medicine might be to improve the health of a patient. However, it may well be extremely difficult to predict all the consequences of the medicine. It may have terrible side-effects. It might not help at all. So what's the doctor to do? Nothing? Roll a dice? Go by a rule of thumb for treating a different illness (leeches, maybe)? Or use all available information to make a decision based on a balance of risks?
The fact the it is not easy to follow the consequences of an action is a problem for anyone trying to act in a utilitarian way. It is not a problem for utilitarianism as a system of morality. After all, we don't say the general theory of relativity is wrong just because it is less easy than Newton's laws of motion...
Today, we see Bentham's theories applied everywhere. A corporation fires 40% of their employees in order to assure jobs to the other 60%.
Again, 60% is very far from 'all', the greatest number. The only instance in which a utilitarian would agree with 40% of a workforce being fired (presuming being fired is a bad thing...) is if the all other options were worse (for example, if other options required at least 50% of the workforce to be fired).
A liberal government reduces taxes, to assure the "happiness" of the majority of the citizens - no matter the minority becomes reduced to extreme misery.
The way many utilitarians weigh pain against pleasure, "happiness" would weigh very little, if at all, against "extreme misery". Depending on the type of utilitarianism, it is very unlikely that a utilitarian government would act in this way.
However, before we discuss ways of weighing pleasure against pain, I think we need to discuss consequentialism is general.