You see different things because different things work. We are all unique. Each flock has its own dynamics based on the personality of the individual members. We have different amounts of room, climates, experiences, tolerances, the list goes on and on. The age and maturity of the chickens involved makes a difference. Sometimes it is as easy as turning them loose together. Sometimes it is a lot more challenging.
If you are going to try just turning them loose you usually need a lot of room. I'm not talking about that 4 square feet in the coop and 10 square feet in the run type of room, I'm talking about enough room to run away from a bully and actually get away, then stay away. I don't know how big your nice big run is. Since you are only talking about five chickens I don't get a great feeling that it is actually that big. The size of the coop is less important as long as they can get outside to the run if they need to.
Integrating immature chickens with mature chickens takes more room than if they are about the same stage of maturity. In your situation I'd look for pullets about the same age as yours, or at least girls that are also laying. That usually works better.
I don't know what you have to work with as far as coop, run, or weather so it's hard to make specific suggestions. So I'll go generic.
most say you need to give them some time to get to know each other without actually being able to touch each other. So, how does one actually do that, practically speaking?
In most cases I think you do need to give them time to get to know each other. Very few people on this forum have the type of room for another approach. From what I see in your post I don't think you have this kind of room.
What you need is a place predator-safe where they can see each other across wire. That could be something made out of chicken wire in a predator-safe coop or in a predator-safe run. The enclosure itself may need to be predator-safe. They need to be protected from weather. What your weather is like will determine what you need. If they are laying eggs you probably want a nest. They don't have to have a roost but giving them a roost higher than the nests might keep them from sleeping and pooping in the nests. And of course they need to eat and drink.
That's all you need that I can think of. How you provide a predator-safe weather -appropriate area where they can see each other depends on what you have to work with. I have a separate coop with its own run across wire from the main run, I use it as a grow-out coop for juveniles. It works great for integration. I have over 3,000 square feet outside where they can roam together. I don't force mine to share a small space until they have shown that they can share a big space. I don't have issues with integration but a lot of people do. Room is not the only key to integration, but I think it is a big part.