Yes, it is possible for most missions to complete the heroic without finishing the mission. But as long as we are not allowed to use Story Mode -> Continue for speedruns we have to beat the mission anyway to advance. The only exploit I can imagine is a mission where it is faster to beat the mission, restart, get the heroic, exit to warrens as opposed to doing both in one go. Cannot think of any missions that fit this criteria though.

In the current version of the game (1.0.3) it is possible to skip to the final mission right after the tutorial due to a bug.

Setup: You need to complete the final mission in any way until the credits are rolling. Press Space Bar (or your equivalent key) or click the continue button to exit the credits. This presumably stores the input of your character talking to the final quest person. After exiting the credits you will enter the main menu. Do not close the game, else the input gets lost.

Using the warp: After doing the setup, select Story Mode -> New Game and start the speedrun. After you complete the first mission and enter the hub world for the first time (Bellafide's room) the mission dialogue for the final archimedes mission should pop up automatically. From there you can access the last mission and end the run as usual. This should also automatically set up the next warp.

Since this bug requires NG+ we might need a new category for this one.