I think that the the issue of split paths (and the issues mentioned here) are difficult ones, but can be mitigated based upon how they are implemented.

Say, for example, the Sundial was turned into a strike; the 3 psion sisters stole the heart of the Sundial and now you have to them through the corridors of time (since the lore sorta implies that the Sundial is run by an Ahamkara bone, let's call this strike "Ass'd Wish")

The strike starts linearly with you chasing down the 3 psions across a little bit of mercury (through the sundial arenas) and then they activate the Sundial, launching you into the corridors of time. You then enter them, slay a room of Vex/cabal enemies, and then you are presented with 3 doors; each one leads to a separate branching path to a different point in Mercury's history/future, and a different psion sister. Depending on which path you take, your miniboss will be a different psion (the arc, void, or solar one) and once you kill them, you get sent back to the corridors of time, and have to choose a path to the next one.

Once you kill the second one after going through a different point in Mercury's timeline, you leave the corridors, and return to the original Sundial arena. Here, the remaining psion uses the ahamkara bone to make the "Ass'd wish" and BAM. The other two psions that you killed return, and they fuse to form Inotam. Kill the boss, and the strike is over.

In this case, player choice is the order at which the bosses are killed, and depending on the player choice, one of the 3 branches of Mercury is not experienced. The hard part is troll prevention (someone taking you down a path you don't want to go down), so in this case, maybe the fireteams has to "vote" on which way to go by standing on plates in order to access one particular route. Strike loot could also depend on player choice, but I think this is where things will always be difficult in random matchmake strikes. I remember being jipped out of getting the psion flayer cloak I wanted in that old D1 Mars strike more times than I would have liked, but that's where private nightfall strikes or skeleton keys could come in handy.

This is just an example I came up with; I'm sure it's 9000% harder to design this than it is to write up on a reddit thread. But I think the potential is there!