Let me provide some technical background, veeeery simplified
Induction machines, from old-fashioned squirrel cage induction machines to more modern torque-starting IMs and hybrids have the big advantage of having a super simple, extremely durable and easily sealable design. Induction machines have stator windings and just a big magnetically conductive segmented rotor. The stator windings induce a magnetic field in the rotor, which in turn pushes off the stator field and generates a torque. Because nothing rotating needs to be electrically coupled and nothing has to be measured (you just put an ac waveform on it and it works) this is great for automotive applications as the design can be safe life for decades. Trains very often use IMs. IMs do lose out on efficiency though, because they necessarily have a LOT of slip (misalignment between rotor and stator windings) which wastes field strength and the induced eddy currents in the rotor cause quite a lot of resistive losses, losses that have to be fed through an already relatively lossy medium. Also, torque is proportional to rotational speed (sort of), so you get very little torque at start-up. That's problematic for a car, and requires either some kind of slipping transmission or fairly large reduction gearing to work in automotive.
Permanent magnet machines (AC, DC, brushed, doesn't really matter) have the inverse characteristics of an IM. Either rotor or stator has permanent magnets, causing a fixed magnetic field to push against. As the permanent magnet side speeds up, it will induce an opposing electromotive force in the windings, reducing the effective field strength and thus torque, up to the maximum free-spinning speed, where back emf is equal to feed voltage. This means that any PM machine has maximum torque at stall and no torque at max speed, linearly. That's great for automotive, because realistically you only need torque at low speeds and the RPM range is typically fine for normal road speeds. The bulletproofness is gone though; you'll either need brushes to power rotor windings (PMDC) or you need to switch at least three poles of stator windings (BLDC/PMAC) out of phase. Also, magnets are pretty expensive, especially rare earth magnets, and you need more of them to get more power out of the motor, as the permanent magnets have a fixed field strength and thus maximum attainable flux linkage. Need more torque? Then you need more magnets. Need more power? Either speed everything up, complicating electronics or increasing brush wear - or more magnets.
So, what do we want in automotive? Well, we want oodles of torque at low speeds so we don't need any gearing, we want a high attainable top speed and still some torque left at that speed and we want everything to be as cheap, light and small as possible. Also, it needs to be safe-life, i.e. we don't ever want to need to replace this part.
What is limiting PM machines in maximum torque and power? Well, the permanent magnets. They're awesome for giving free field strength, but with a limit of about 1T we can't scale flux linkage much more. However, electromagnets can create... well, any field strength, up to infinity. We also don't want brushes, because they're not safe life, so all the electromagnets have to be in the stator, the rotor has to be passive.
This is what switched reluctance motors are. Like BLDCs, they only have stator coils, and like induction motors, the rotor is just a passive piece of magnetically conductive material. They're different from IMs in that the rotor doesn't generate an induced magnetic field, but that the stator poles and rotor poles are just ever so slightly misaligned, so that the magnetic field produced by the stator is conducted through the rotor to produce a torque without slipping. This requires extremely precise timing of the stator coil cycles.
The higher field strength means much higher field strength and thus much higher torque can be generated, which in turn means higher specific power (smaller motor / more power). The nature of the rotor means you have a PM-type torque response and the only real limit on RPM is how fast you can reliably detect rotor position and switch the coils. The 'perfect' electric motor.
Well, until fairly recently that was totally not the case. A typical SRM has many poles, at least 6 but usually 18-36 for a rotary SRM. You need these many, because along with the very precise timing you also get a very uneven torque distribution over the rotation of the rotor. It's an oversized (non-PM) stepper motor, so there are very pronounced steps in its rotational motion. This in turn leads to high mechanical loads (vibration, bearing wear, etc.). Also, having to switch that many poles in very odd sequences and current profiles is an electrically challenging task, leading to relatively unreliable motor drivers. This, however, has been mostly solved these days through the use of all solid-state precision sensing and switching topologies. Much of this work stems from companies like ASML, who use linear SRMs exclusively for precision (and high-speed!) chip manufacturing machines.
But, these days the electronics can all be solid-state and safe-life too. So even though there is a lot of complexity in these motors which should usually be a red flag for safety and reliability, it's all solid state and if designed properly can be safe life. So far still mostly in theory, because you are correct: all implementations for automotive use right now, mostly used in precision vehicle control like robotic pickups, are noticeably noisier and lower power than BLDC counterparts. I am not aware of any big brand committing to SRMs in a production EV at the moment. There is also not a shortage of rare earths for a decently long time to come, so I assume they'll just keep making PM BLDCs for a while.