So its an artificial gating system that results in breaking the social aspect of the game. Seems dumb.
Also, the tower is heavily instanced, so you aren't going to see the same people there very often. And either way, how do you get to know them when its impossible to communicate?
When I first started playing, I would just invite people from the tower to my fireteam. Or if I was out on a mission by myself, you can see other people in the area and I would invite them to my fireteam as well. If they accepted you would then play the mission with them and at the end of it, I would decide whether it was someone I wanted to play with again and just add them to my friend list.
The why the friend list works, you don't need the other person to accept. All you have to do is just add them to the list and you can see them whenever they are on again and see what missions they are currently doing. So I used to have like 20-30 people on my list and once I logged in, I would see if anyone was already on a mission that I wanted to do and if their team wasn't full just ask them if they mind if I joined. More often than not if they were doing a mission solo, they would be fine. The same thing also happened to me where I may be on a mission and someone I did a mission or strike with before would ask me if they could jump in.
That's ultimately how I built my network of people so that by the time Raids came out, I already had a group of people I had played with quite a bit. Of course, this was back when everyone was building up to level 20 as this was when the game first came out. For the guys that came to the game late it was probably harder because you had a lot of high level guys running around already and the population of people at the low levels trying to build up their character was much smaller.
