cosmology - Is the Universe really expanding at an increasing rate?

I don't take this at face value because we should expect more distant
objects to have higher observed speeds and therefore higher observed
red-shifts.

That's true. That was the original Hubble discovery - the farther away things were, the faster they were moving away from us.

Here's why. Let's start with a model where the Universe expanded very
fast early on, but has been slowing down ever since due to gravity, as
one would normally expect.

Yes - that's what everybody thought following Hubble's discovery.

Remember that, the farther away a cosmic object is, the farther back
in the past we are observing it. An object 1,000 light years away, if
it's light is reaching us now, is being observed in its state that
existed 1,000 years ago. We are effectively looking through a time
machine.

This is not lost on Astrophysicists.

So if we observe a more distant object, we're observing an older state
of that object. Therefore, we are observing it at a time when the
Universe was expanding faster than it is now, so it has higher
red-shifts.

OK, 2 points. 2nd point first. The red-shift has to do with relative velocity, not speeding up or slowing down. Something can be more red-shifted and slowing down and something can be less red-shifted and speeding up, especially since the acceleration/deceleration is comparatively slow compared to the relative velocity.

and other point - lets keep in mind, we don't know what a galaxy 3 billion light years away is doing now. We can guess and we can run models, but we can only see what it's doing 3 billion years ago.

And isn't that what we observe today? The more distant the galaxy, the
higher its red-shift? This is not inconsistent with a "normal" model
where the expansion is slowing down due to gravity.

Yes, the more distant the galaxy the higher it's red-shift. But no, that's not inconsistent with expansion. That's what you'd see, expansion or contraction, because red-shift is just relative velocity.

What am I missing? Why are scientists trying to explain such things
with weird dark matter and dark energy that otherwise have never been
detected or found evidence of and aren't needed for any other model,
and in fact get in the way of our models of physics and quantum
dynamics?

A lot of these ideas are confusing. They're confusing to scientists too, especially when they were first discovered - so you're not alone.

Dark matter was observed because galaxies were behaving strangely. The stars in the outer arms of the galaxy were observed to be moving much too fast and faster than the stars more towards the middle of the galaxy and that made no sense. The galaxies also weighed too much and the only way to explain this was extra mass in kind of a halo around the galaxy, but this extra mass, also, didn't interact with electromagnetic waves like the mass here on earth does - so they called this extra mass (and there's a lot of it, more than there is regular mass), but since it's invisible, they called it "dark matter" and it's not dark like dirt or coal, it's dark as in - invisible. It's completely transparent to light, but it has mass and they still don't know what it is. They have some OK theories, but nothing definite.

Now, dark energy - think about the big bang and all matter flying apart - the galaxies twice as far are moving away twice as fast, BUT, as you said, because of gravity, we should see the galaxies that are twice as far moving away more than twice as fast, cause the nearer the galaxy, the more time it's had to slow down - aha, they thought, if we can compare the speed of the galaxies 4 billion light years away to the speed of the galaxies 2 billion light years away to the speed 1 billion light years - etc, etc and measure it all carefully, we can measure the rate at which gravity is slowing down the universe. - that makes sense right.

And with careful measurement of Type 1A supernovas, which temporarily outshine entire galaxies - with remarkable consistency (what they call a standard candle - a very bright standard candle, but a standard candle all the same) - with that, they thought they could measure the gravitational slow down of expansion - exactly what you're talking about.

The problem was, the measurements told them the opposite of what they expected to find. The measurements told them that the galaxies 2 billion light years away were traveling slightly more than half as fast as the galaxies 4 billion light years away, and so on. They checked this, cause it had to be wrong, then they re-checked it, and re-checked again and the only conclusion was, stuff out there is speeding up, not slowing down - cause that's what the telescopes tell us.

Dark energy wasn't a hair-brained scheme that mad scientists thunk up. It was an observed reality that nobody expected (well, cept just maybe for Einstein and his cosmological constant, but that's another story).

Dark energy's just a name anyway. They have to call it something, even if they're not sure what it is or how it works.