Whats better a 2 way speaker or a 4 way speaker

The legacy speakers are a good indication multiple drivers can integrate well with better than subwoofer bass. Better in that a sub is difficult to time-align with the rest of the drivers and they give one-note bass, the leading edge of the note is dulled, with less transient snap, missing too much deep information detail.
Legacy is not the only company to use many drivers successfully. Vandersteen, psb, Paradigm, Sonus Faber, and the lust goes on. Jallen

If you're comparing mini monitors to towers, your friend is generally right on most counts.

2-ways have the advantage of less phase shift caused by large value inductors required for the lower crossover frequencies. 2-ways lose some efficiency with a L-R network for diffraction loss but that's no different than padding down a midrange/tweeter to match a less efficient woofer. That's where the .5 comes in for 2.5-ways but that usually still has that big coil for parallel drivers.

Having recently completed a 4 foot tall, 2.5 way, series/parallel with 4 XBL^ woofers per side, that part about the fuller sound is only a generalization. And they're an easy load with high efficiency. There are some 7' tall 2-ways with dozens of drivers and some of those can fill a stadium.

Your friend is also wrong about the subs. A well designed and placed subwoofer can actually be easier to integrate into the room and subwoofer drivers can do deeper bass in smaller cabinets that's beyond capable with wide range woofers. The problem is that the satellites usually don't have a high pass and still contribute distortion as they try to do what they are poorly suited.

For me personally a 2 way monitor with sub works best. When you get the sub set up properly to just add to what the monitors are physically not capable of reproducing you will be in heaven. For me timing with the sub will make or break my statement.

That being said it has to do with your room size too. My room is 26.5' X 12' BUT the speakers have to be on the long wall which I would consider somewhat near field listening. Floor standers sound just OK in my room. They are either too forward or have a vertical sound stage that to me is unrealistic. In a properly sized room the floor standers will be better and probably be fine without a sub.

So in the end what I'm saying is you both could be right depending on the room.

The quality of sound from a speaker is not a function of the number of drivers but as you have read it is the cumulative success of engineering all of the elements that go into the final product.
Some people prefer the 2 way speaker and others like line arrays. You just can't generalize so both of you are right and wrong. Right because there are fantastic 2 ways like the Merlin floorstander but wrong because there are fantastic multi driver line arrays such as the Infinity IRS V . I heard a pair of both of these on multiple occasions at friends homes. You should look at the McIntosh and Genesis speakers as other examples of line arrays.
My own "big rig" speakers are 4 drivers Focals, which I think sound fantastic of course after getting the right electronics for them. My living room system speaker are 3 way front firing VR-2s which have a mid/tweeter ambience driver firing on the backs.

To begin with, why were 3 ways left out of the discussion?
There have also been 2 ways which are capable of considerable extension in the bass, and 3 and 4 ways that required a subwoofer for some listeners. In the end, it's the same old answer. Listen, and trust your ears and preferences.

Depends on what the 2 way is trying to do. If the 2 way is trying to reach the depths, in all likelihood it will compromise the midrange. If it is properly designed and rolled off no problem.

More drivers means more complicated crossovers which, again depending on design could be a disadvantage. As Pbnaudio says, they are designed for different purposes.

There's good 2 way speakers and there's good 4 way speakers, a 4 way by nature requires a more complex crossover than does a 2 way, but a 4 way crossover does not have to "work harder" than a 2 way crossover, but usually your amplifier has to work a little harder with the more complex 4 way crossover.

Your friend is correct about extension and loudness.

Cramming a 4 way tower into a small bedroom is not a good idea neither is a small two way in a large room

Good listening