Hmmnnn... I would say that if Archmage or Heirophant were the first path selected, then yes. Dual Path only says that you get the additional 1st tier ability and can pick from either list of path abilities. Since it doesn't say you give up any of the first path abilities, nor do you gain anything that is not specified, then dual path archmage / champion gets mythic spells, but champion / archmage does not (as written)
However, since the other classes don't have anything that compares to Mythic Spells that seems kind of wonky to me. it's not like there is a non - path champion ability that allows you to burn a mythic point to auto crit (for instance) x/day that would be comparable to the mythic spell feature that you would have to choose between to determing starting path...
That might actually be a way to balance it, though. Give each path a limited use mythic point burn to improve a class feature for 1 round or one use. Just off the top of my head:
Champion: Mythic Strike: auto-crit (and auto confirm) ignoring all dr a number of times per day equal to your mythic tier /2
Guardian: Mythic Dodge: you can completely ignore the effects of one attack (whether physical or magical) or you can grant this ability to any ally in sight a number of times per day equal to your mythic tier /2.
Trickster: Mythic Feint: You can spend 1 mythic point to automatically succeed on a feint maneuver as a free action a number of times per day equal to your mythic tier /2.
Marshal: Mythic Inspiration: You and all of your allies can ignore a targets SR and DR for their next attack a number of times per day equal to your mythic tier /2.
Thoughts?