Are you 100% sure that the other 3 players survived the elevator ride down then?
They would only do that then, if they already completed the full Phantom program and including making the ride down the elevator earlier.
What you describe is what happens when the group of minimum 4 players do not complete all the requirements to enter the Phantom vault room at the bottom of the elevator ride down. Full Phantom graduated players can always make the ride down no matter what. But new players that want to graduate needs to be fully dressed up for it as you describe you try to be. And in total you need minimum to be 4 players in total to make up the full combination required. Where as said above, already full Phantom players can join and stand in as 'joker' for whatever combination of dog tag(s) the other 'newbies' might miss to make them all 4. But all newbies will die at bottom if not all requirements are fulfilled.
And it appears you have several 'check points' during the elevator ride going down:
1. Phantom dog tag check is first. Those not having that attached die early in the ride!
2. Next check during elevator ride is for the Phantom soldier cameo! All not having equipped their soldier with this will die.
3. Third control during elevator ride is for the weapon's Phantom cameo. This is the last control point before the doors open up again.
I seem to recall that the ride takes around 30 seconds in total, and you will have a couple of times when the lights switches ON/OFF and other soldiers may appear to die during those control points. So using this, you might be able to identify when on the elevator ride you die and thereby also the reason why you die, per my description above.
Best of luck with it - its still something very special to enter the bow room down there - so lots of fun and worth the effort!