This argument is so irksome

There's truth on either side, and too much dogma always ruins a good discussion.
Let's take it for granted that it is fundamentally a good thing to learn how things in the world work (if we don't accept this then, fine, but let's all go and live in caves with nothing more advanced than fire and the wheel).
The entire fundamentals of neuroscience (I'm going to stick to this because it's my discipline so I know about it) were established, tested and verified by animal experiments. Just a couple of basic examples here.
Being able to examine how non-human brains respond to stimuli before the development of non-invasive techniques was crucial to the development of understanding of how neurons work and how brains are wired up.
The first experiments on nerves were done on giant squid axons, and these were not in the least bit misleading, the basics established there are basically 100% correct for all nerves. OK, some squids died, I really don't think there are many people who mind this, since they're not especially fluffy or cute.
Reflexes were mostly investigated in aplysia (sea hares). Again a process which required invasive monitoring of nerve fibres to measure what was going on and monitor the transmission of information through neural assemblies. Again, what was discovered is absolutely correct and applicable to non-sea slugs.
I could go on, but in summary everything that we know about nerves had to come from animal research because looking at dead people's brains wouldn't have been any help whatsoever. And there aren't many people with iopen skulls for us to investigate. Although, if anyone's up for it, there are no pain receptors in the brain so you wouldn't feel a thing (that is after your skull's been sawn open).
So, knowing how a neuron works to receive and transmit signals is one thing, and now we want to know more about how the brain works. It turns out that since it's a hugely complex system we're unable to model it accurately- even modeling 1 single nerve's interactions with all those it's connected to is pretty much impossible since the system is in a constant state of flux. Therefore, we need a system to examine which is representative.
It is perfectly legitimate to attempt to gain fundamental information from looking at the way animals' brains work to do basic processes, because evolution would suggest that there is a lot in common between species' brains at fundamental levels of processing. And this is in fact borne out by reality.
Basically, we can put electrodes into the brains of animals and look at what they do with inputs, how they represent information and so on, and it gives us a pretty accurate idea of how the human brain can do the same thing. It is unlikely to mislead unless it is overstretched, so maybe looking at the brain of a guinea pig in order to understand how humans process sentences is going to be silly, but looking at a mouse in order to see how brains generally represent click sounds is far less unreasonable.
There is no alternative, unless people suddenly start to have a penchant for having their skulls opened and electrodes whacked in their brains for several hours at a stretch, with potentially damaging consequences.
At present non-invasive techniques for looking at the brain have a resolution of thousands of neurons at best. If we ever want to be able to model the brain (in order to obviate the need for animal testing in neuroscience/psychology) then we have to continue to do variations of single-cell recordings using arrays of electrodes actually plugged into the brain.
There will be new non-invasive techniques, but we will need to validate them in order to be sure they work, which will require detailed information, which can only be obtained from invasive research...
It's a catch 22.
Some animal testing is flawed. Some isn't. To say that it is all fundamentally flawed and exists only because of a conspiracy is fallacious and wrong-headed.
There are still areas where it is the only way to get at information.
Ultimately, we all want to see an end to animal testing, whether for moral, ethical or financial reasons (or all!), but the only way this will happen is if we gather enough information about fundamental biological processes. Once we've got those we can work on making computer simulations. And even then we'd still have to test the safety of new drugs on something because how could you know for sure that the models were totally representative?
The complexity of human biology is such that we could never really be sure.
edited because Aplysia are sea hares not sea slugs (FWIW)