I understand the advantages of using a rectifier-inverter system to drive an AC motor instead of simply plugging it into mains power, as it allows much better control of its speed and performance; but what I don't understand is: since the original AC power has to be converted to DC in order to feed the inverter circuit, why is this DC not directly sent to a DC motor, instead of converting it back to AC and then sending it to an AC motor?
DC motors have effectively one variable: how much power are you feeding the motor? AC motors have two variables: power, and frequency. I'm not an expert in motors, but I'd expect that AC motors would thus allow independent control of speed and torque, while DC motors do not. Directional control is also a concern. An AC motor's direction can be controlled by the rotational direction of the power being fed to it. A DC motor's direction isn't so easily controlled.
More broadly, all motors operate because there's a rotating magnetic field somewhere. That rotation is either generated inside the motor (self-commutating) or because the power feed to the motor is itself rotating (externally commutated). DC motors must be self-commutating; DC is definitionally not rotating.
How do you achieve commutation inside the motor? Typically, either there are brushes, or there's an inverter built into the motor. Brushes wear out, and I suspect have other disadvantages. And if you're going to build an inverter into the motor, why not put it outside the motor and get better control of it?
Because AC motors are generally much more efficient than DC motors, and since they don't require electrical contacts to the rotor, are more reliable as well.
Remember, a BLDC motor is really an AC motor with the drive circuitry built-in. At higher power levels, it makes sense to separate the control and drive circuitry from the motor itself.
Also, motors with permanent magnet (PM) rotors have limited power-handling capability. At higher power levels, AC induction motors are used, even in electric vehicles.
With many kinds of AC motors, the rate of rotation will be strongly correlated with the frequency of the driving current. In many cases, the rotational speed in revolutions per second will either be an exact fraction of the drive frequency in cycles per second (e.g. 1/3), or else an exact fraction minus a certain amount of "slip" which depends upon the drive voltage. While it may be possible to control the speed of some AC motors by varying the drive voltage and thus allowing varying amounts of slippage, it's more efficient to vary the drive frequency and try to minimize slippage.
Note also that nearly all motors that are capable of doing a non-trivial amount of work require that the polarity of current in some of the coils be periodically switched. This is just as true of DC motors as AC ones. Most DC motors use a mechanical commutator and brushes to perform such switching; these tend to have a limited useful lifetime before requiring service or replacement. Some use electronics to switch the actual motor current, but that essentially turns them into an "inverter-plus-AC-motor" combination.
There could be lots of reasons. The most obvious is that the brushes in PMDC motors wear out and need to be replaced after 2000-5000 hours, depending on the environment. Whereas AC motors (both induction and PMSM aka brushless motors aka BLDC motors) can last 20,000 hours. So if maintenance-free operation is important, you might want an AC motor.
Second, if you are doing any sort of speed or torque control you aren't going to just have DC for a DC motor. You are going to have PWM DC. And once you have the electronics to do that, it isn't that much different to go to PWM AC.
Third, a lot of modern induction motor and PMSM controls operate using a technique called field oriented control. This type of control allows you to control both the operate your motor smoothly at low speed and high speed and it gives you independent control over your torque and your magnetizing field. You can't do this with a PMDC control because your brushes/commutator mechanically align the field. So if that is important to you, you might choose an AC over a DC motor.
AC motors are more reliable than DC motors. DC motors produce output power from the current flowing in the armature. The DC motor transfers the current to the armature with commutator and brushes. The electrical inductance of the armature causes arcing as each brush breaks connection from each successive contact bar of the armature. This pits the armature and brushes, making them rough. The roughness wears both armature and brushes. When AC motors use electromagnet rotors, current connects to the rotor with slip rings and brushes. There is no switching at the brushes on slip rings. This avoids the pitting from arcing suffered by DC motors. Slip rings and brushes last many times longer than DC motor brushes and commutators. Most AC motors operate without brushes and slip rings by using inductive coupling, hysteresis, or permanent magnets in the rotors. Service life of brushless motors might be limited only by bearing life.
AC motors can be more controllable than DC motors. DC motor controllers can change the magnetic field from the stator or the voltage or current applied to the armature. AC motor controllers can change stator voltage, current, frequency, or phase, or rotor current. Some AC motors can change the number of magnetic poles on the stator. This makes AC motors able to efficiently convert electricity to motive power over a wider range of operating speeds than DC motors of equivalent power levels.
Another aspect that I don't see mentioned is that a 3-phase AC motor fed with pure sine wave input produces uniform torque through all 360 degrees of rotation. A simple DC motor will experience torque variation as each rotor pole rotates past its counterpart stator pole. This can be an important consideration in, for example, precision machining.