4e monsters didn't get extra turns generally. A few did, but it wasn't standard practice. Now in late 4e "instinctive" actions became a thing and they allowed a specific action at your initiative +10, but that is not a full turn. Unless you are talking about action points which I always forget about (and did in play too) and can't remember how exactly they worked.
It was more common in later 4e monsters to have monsters go at initiative 20, 10, etc. Now often these monsters could only do a standard action at that initiative (not a minor), but you still got the notion of the monster going multiple times in a round. I actually just looked back at a few myself to check myself. You are correct taht not all solos used that mechanic, which again I think plays into 4e;s strength....we saw more different designs of monsters to shake things up...whereas legendary monsters all fit a very predictable format.
Beyond that, here are some of the things I liked about that model:
1) Solos that went multiple times had multiple chance to get over conditions and effects. If I knock out a legendaries action....it loses everything. Solos could only lose a portion. 4e solos also had big saving throw bonuses to get over conditions faster.
2) Action Points were very simple but created a BIG boost in threat if used properly. The fact that at any point I could just slap in another big action that 2nd its needed right on top of the last one can change the battlefield. Its a sword of damocles held over the players that I can use when the iron is hot.
Or...I often used this with what I called the "recurrent villain power". I would spend them as a retreat option to get the villain away quickly, live another day and go after the party again.
3) Simplicity: I think the solo monster model is just easier on the stat-block. I have X options, and I have Y turns per round. No "well this option takes 2 legendary actions, and this one takes 1, and this action is only when the monster truly goes not for a legendary, etc".