And if the wave function collapse when measured wouldn't the fact of looking at the screen to see the interference pattern collapse the wave functions of the electrons and make the interference pattern disappear. Why does it only disappear when we specifically detect through which slit the electrons are going through? Why doesn't looking at the screen count as an observation/measurement?
In collapse interpretations, when the wavefunction is observed it collapses to a single localised peak (like a particle), but then thereafter starts spreading out like a wave again.
So if you observe the wavefunction as it passes through the slits, it collapses to a point passing through one slit, then this single-peak wavefunction spreads out again as it heads for the screen. At the screen it is observed again, collapsing to one point on the screen, which has the distribution of a single-slit spread (i.e. no double-slit interference pattern).
If you don't collapse the wavefunction at the slits, the wave passes through both slits at once, and being a linear differential equation, the two-slit pattern is the sum of two one-slit patterns, which results in an interference pattern. At the screen, the wavefunction with the two-slit interference pattern collapses to a single point.
Looking at the screen does count as an observation/measurement in both cases. But the wavefunctions being collapsed are different - either the one-slit pattern if you have previously collapsed it by observing it at the slits, or the two-slit pattern if you haven't.
I know it's common to say that when states are in superposition the electron is in all of them at the same time but I thought that was a vague way of explaining it when in reality what's going on is that there is a probability for each state and we don't know which one it is until we measure (so we say it's all of them).
It is a vague way of explaining what is going on, but rather than being a case of a particle of uncertain position that we are incorrectly/fuzzily treating as a probability wave, it is actually a wave that we are incorrectly/fuzzily treating as a particle being in many places at once.
In the collapse interpretations, it is not a particle until you observe it. It is not a probability, either, until you come to actually observe it. Probabilities are real numbers between 0 and 1. Probabilities aren't complex numbers which can destructively interfere with one another. The wavefunction behaves as a complex wave, which when observed (and only then), gives rise to an associated probability of observing it at each point.
As you say, the single electron must go through both slits at once, or else there would be no interference. An electron passing through only one slit would have no way of knowing about the other; whether it was open or closed. Waves can do this. Particles can't.
The tricky part is interpreting what happens when we observe it. The wavefunction hits the entire screen at once, and interacts (lights up) at only one place on the screen. The rest of the screen somehow knows not to respond, even if there is no way for a signal to get there from the one place in time. Collapse is non-local - the transition from the spread-out wavefunction to the single-peaked, highly localised wavefunction appears to propagate faster than light, and hence, in a relativistic universe, backwards in time for some moving observers. It would be understandable if you rejected this option as logically impossible too.
There is an alternative interpretation that retains locality, which requires you to treat the observer as a wavefunction too. Then when the observer interacts with the electron, the wavefunctions become 'correlated'. Just as the spread-out electron is a sum of localised electrons at each point of the screen, so the observer becomes a sum of observers, each observing the electron at a different point. Just as different parts of the electron's wavefunction don't interact with one another (there is no electromagnetic repulsion between different parts of the wavefunction, for example), so the summed observers don't interact with one another. The physics is deterministic (all outcomes happen, every time) and local (the observers seeing it hit one part of the screen can never perceive any observers seeing it hit another part, even if they all go to the same place later). It only appears random and non-local because we can only see our own small part of the bigger picture. There is no collapse, only correlation. There are no particles, only waves. One wave looks like a particle to another wave when they are perfectly correlated; when they move in sync. Not everybody likes that idea, though.