Decoyrebbe:
You wrote: “why in melochim the posuk tells us that Eliyahu Hanavi runs to mount Sinai to escape people chasing him”
It seems that the trek to Har Choreiv was an instruction from Hashem, not the escape route that Eliyahu chose. His intention was to get into the desert where Ach’av wouldn’t find him. This is not stated openly in the psukim there, but the context suggests that. I might offer that the reason why to Har Sinai was that the message of קולות וברקים of מתן תורה has a parallel to the רוח ואש of the experience Eliyahu would have there. Both carried the message that the drama was not a permanent thing, and that קנאות has only a limited place. Since Eliyahu was insistent that this mode continue, Hashem instructed him to anoint his successor, אלישע.