My friends and I managed to find a way to skip the first two missions of Taken War: Earth, Venus, and Mars to get access to the Road to King's fall mission. This will make it easier for your alts or even your main if you haven't done it to get access to the quest for the raid to get a 310 artifact quicker.
Notice: We don't know if there is a way to make this method even shorter or if the order even matters, but this is the way we did it and it works.
Step 1: Make sure you have Taken War: Earth and Venus in your quest log. (Just need to have it even if nothing was done in them) Edit: (A friend just confirmed that you don't even need these in your quest log but if you have them just move on to the next step)
Step 2: Join a friend's game that has done Phobos "Fear's Embrace" and join him/her. Finish that mission. (We joined a friend that already completed the entire questline and had access to all the missions)
Step 3: Go back to the tower and you should have access to "Blighted Worlds" from Eris now.
Step 4: Go join your friend again and complete "Kings of Decay" on Earth which is the last mission on Taken War: Earth.
Step 5: Go join your friend again and complete "Entropy's Pinnacle" on Venus which is the last mission on Taken War: Venus.
Step 6: Go to the tower and you should now have access to the first part of Road to King's Fall from Eris.
Step 7: Complete "Summoner's Circle" on the Dreadnaught and then go turn it in to Eris for the final raid quest for the 310 Artifact.
That's it, you're done in just 4 missions on any alts you may still have left to get it on. Just to reiterate, this is the way my friends and I did it. There may be other ways to achieve the same effect. Good luck out there guardians.
Edit: Doing this gives you "Road to King's Fall" which is the raid quest to beat the raid bosses to receive your 310 artifact.
Edit2: A lot of people seem to be confusing
The Road to King’s Fall quest with
The Old Hunger quest. The Old Hunger quest is for the Touch of Malice questline and doesn’t reward you with a 310 artifact.
Also, skipping the taken war quests doesn’t bug out the game or anything and you can always go back and redo the first quests of each questline for the legendary marks if you want to.