I prefer a well-designed 2 way to a well designed three way. Too me, there's just too much added complication to a 3-way, which always leads to added circuitry or more complicated design parameters intended to compensate for the problems inherent in adding a 3rd driver.
That's not to say a good 3-way couldn't sound better than a good 2-way, I just feel you add cost.
If cost is no object and ultimate performance is sought, then it's quite possible a 3-way could be superior.
There are cases, however, where a 3-way could offer similar performance to a 2-way speaker and be less expensive. Much would depend on your goals for the speaker, and the environment the speaker will be placed in. My own bias is to keep things simple.

Not all 3-ways split the frequencies as you describe. Some of the best speakers I've heard use 2 focused super tweeters and a full range woofer that handles everything below 8 kHz.

I wouldn't recommend shopping for a speaker based on its driver complement unless you have good reason.

My experience with speakers below $3000 or so is that a good 2-way will have fewer issues, but this is a broad generalization.