So If I was to offer a solution to the no matchmaking in game, while preserving the challenge of the raid itself, it would have to be in game. I think finding people on forums and Clans is good to start, but it could mean nothing when you actually play together.
So my solution would be placed in the tower. And probably centred around the Postmaster.
In short.....SPEED DATING!
So at the tower's postmaster, people who would like to start a raid or make up a team for whatever drop in a request, along with everyone else already in the fire team, to start a team that everyone else can see, regardless of if they are in the same instance. This set up would then show the stats of the players currently in the "Lobby" so that people get an idea of the group make up, maybe a motto as well. Then people who accept this can be dropped into a voice chat session and they can determine if they can work together. Once all slots are filled they can start the instance.
There are holes in this idea I'm sure. But if you can think of a way to improve it I'm all ears.
Also look up my Clan: The Buzzards if you like and are still looking for a raid team.
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ps4 clan?