Right reasons for using a voltage regulator for DC motors
It is important to consider that the first line of the quoted material contains the words "for the purposes of this lesson." The explanation that follows may be adequate for that purpose, but it is overly simplified for a general understanding. The phrase "As the resistance of a motor goes down" uses the term "resistance" to describe the combination of the motor's resistance and reverse electromagnetic force or "back EMF." The reason for motor speed decrease is not mentioned. The reason would probably be increased load torque. Also not covered is the performance of the motor from the time it is switched on until it reaches a stable operating speed,

In general, rather than a voltage regulator, it is more desirable to use and electronic speed controller (ESC) with a DC motor, and brushless DC motors can not function without an ESC. If an ESC is used, the ESC performs the voltage regulation in a manner that is more suitable for a motor, and an additional voltage regulator is not necessary. The quoted source may cover those points later in the lesson or in subsequent lessons.

Re: Comment &amp; Link to Lesson

The lesson is a crude oversimplification. The motor used is very tiny when viewed in the universe of motors that includes motors propelling electric vehicles, ocean-going ships and industrial machinery. A brushed DC motor should be modeled as a resistor in series with a generator with polarity opposing the supply voltage. The voltage (back EMF) generated by that generator is proportional to motor speed. After accelerating to the maximum speed with no load, the back EMF is close to matching the supply voltage. Any load tends to reduce the motor speed  and the EMF allowing increased current so that the current multiplied by EMF is equip to the mechanical power produced. The series resistance of an efficient motor may dissipate only 10 or 20 percent of the power taken from the supply with the remainder converted to mechanical power.

Find a lesson that uses the resistor - back EMF model. The one you have found does a terrible injustice to electrical engineering.