Current Draw vs Voltage
In reality, the current doesn't increase without limit.

You are most likely dealing with an ideal voltage source.  An ideal voltage source can deliver as much current as needed to keep the voltage constant no matter how much current the load draws.

Any real voltage source cannot supply endless current.

A real voltage source (much like your water faucet) can only deliver a certain amount of current.

You can view a real voltage source as an ideal voltage source in series with a resistor.  This limits the maximum current that the voltage source can deliver.

Modeling a voltage source that way is often useful.  All real voltage sources have a limit to how much current they can deliver.  That limit acts much like a resistor.

You can prove it to yourself quite easily.

Find a small motor that runs on three volts.  Use two AA cells in series to power.  It will run just fine.

Now, connect the same motor to a CR2032 cell.  These have a nominal voltage of 3V.

The motor will not run when connected to the CR2032.  The little cell cannot provide the needed current.

If you measure the voltage from the battery while the motor is connected, you will see that it is far less than 3V.  The internal resistance of the CR2032 and the (low) resistance of the motor form a voltage divider.