I think a lot of the issues could be addressed by adding filters to the matchmaking process... Team Size, Min Level, Mics, etc... You'd only need a few dimensions to select into a compatable group that fits what you're looking for.
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Probably, but it could be a step too complicated too. I think just a simple "Use matchmaking" prompt could suffice. If you answer yes, you will be added to the matchmaking similar to the regular strikes. Answer no and you will go straight into the strike with your fireteam, without adding randoms if you are less than three in the team, just like it is now.