Why does most of the internet say LEDs will pull too much current without a resistor, but it seems like the resistor is there to bring the voltage down, but the LED pulls the current it needs?
For those who just want an LED indicator circuit without getting mired in excessive detail, treating the LED as if it has a pre-programmed voltage and current and computing the resistor value from that is more than enough detail for many. And it works fine enough for LED indicators using a DC supply.
Either algebra/calculus or geometry/pictures helps gain stronger insights. And there is no single best way to go here, either, as the better approach usually depends on the ways one prefers to think about the world in quantitative ways. For some, that's found in symbolic statements. For others, that's found in graphs or geometric presentations. For still others, both are wanted.
A symbolic way, without an added resistor and for smaller currents, is \$I_{_\text{LED}}\approx A\cdot\exp\left(B\cdot V_{_\text{LED}}\right)\$. Constants \$A\$ and \$B\$ characterize the specific details and are, themselves, functions of the LED chip temperature, among other things. For higher currents, various impedances start to matter (and eventually dominate.) These can be bulked together into a single term, \$R_{_\text{LED}}\$. Then the more detailed equation that covers more territory (wider current operating range) is \$I_{_\text{LED}}\approx A\cdot\exp\left(B\cdot \left[V_{_\text{LED}}-R_{_\text{LED}}\cdot I_{_\text{LED}}\right]\right)\$. Note that now \$I_{_\text{LED}}\$ appears on both sides and solving that requires a diversion into product-logs. But an iterative approach also works. When adding an external resistor, this resistor is just added to \$R_{_\text{LED}}\$ so the new equation is then \$I_{_\text{LED}}\approx A\cdot\exp\left(B\cdot \left[V_{_\text{SUPPLY}}-\left(R_{_\text{LED}}+R\right)\cdot I_{_\text{LED}}\right]\right)\$. That's not any worse than before. But it's still messy. (Note that I changed \$V_{_\text{LED}}\$ to \$V_{_\text{SUPPLY}}\$ because we're now talking about the external supply rail and not the LED terminal voltage.)
The above doesn't provide a lot of insight for most. You can kind of see that the series resistance drops some of the supply voltage. This fits under a general topic called negative feedback (NFB) and when one is tuned into such things then the equation makes some sense. Let's switch over to drawing some pictures to see how this bulk resistance-style NFB works. To do that, let's plot these two equations:
\$y=A\cdot\exp\left(B\cdot x\right)\$
\$y=A\cdot\exp\left(B\cdot \left[x-C\cdot y\right]\right)\$
Above, \$C\$ is the NFB factor (the added resistor), which feeds some of the output back to the input to reduce its impact. I'll pick some values: \$A=3\times 10^{-15}\$, \$B=9.6\$, and \$C=100\$ and plot a result.
Before I do that, one more equation which is the exact solution equation for the 2nd equation above, using those constants I just mentioned:
\$y=\frac1{960}\cdot\operatorname{productlog}\left(2.88\times 10^{-12}\cdot\exp\left[9.6\cdot x\right]\right)\$
Now, here's the plot (performed using Wolfram Alpha's website):
The exponential curve (no R) is the LED without a resistor providing NFB drawn from the 1st equation above. The 2nd equation could be plotted, parametrically, but instead I used the 3rd equation to plot the Exact line, where a resistor has been added to provide NFB. Note that the resistor tends to linearize the exponential curve, so that slight increases in voltage mean slight increases in current (instead of rapid increases.)
I've added what's called a load line for the resistor. (These were in more common use before calculators and computers became so readily available.) Note that this line intersects the exponential LED curve at a certain point. By finding this intersection, one finds out where the LED current will wind up when adding a current limit resistor.
The load line is easy to draw. On the left, on the y-axis, mark a point with \$V_{_\text{LED}}=0\:\text{V}\$ (can't happen, but assume it for now.) In this case, all the supply voltage (call it \$5\:\text{V}\$) is across the resistor. So the LED current would be: \$\frac{5\:\text{V}-0\:\text{V}}{R=100\:\Omega}=50\:\text{mA}\$. On the right, on the x-axis, mark another point with \$V_{_\text{LED}}=5\:\text{V}\$ (also can't happen.) Now, all of the supply voltage is across the LED and there's nothing left over for the resistor, so \$\frac{5\:\text{V}-5\:\text{V}}{R=100\:\Omega}=0\:\text{mA}\$. Draw a line between these two points and see where that line intersects the unlimited LED I/V curve. That's the operating point!
Again, note that the Exact calculation that includes the NFB and solves for the LED current treats the x-axis as the supply voltage axis instead of the LED voltage axis. But please see that right at \$5\:\text{V}\$ for the supply voltage that curve just touches the red horizontal line, right at the exact same current as did the load line intersection. This is why a load line is so nice. No nasty equation solutions to work out. Just draw and it shows you the right result.
Regardless of how you do it, super-math mode or load-line graphs, the above picture should get the point across. There's much better control and predictability when a resistor is added -- good enough for most indicator light uses, anyway.