I believe an Heir’s purpose is to “be provided with their aspect/provide their aspect.” An heir is one who inherits things, taking things for themselves.(See: Mituna taking in the destructive force of the “Doom” in his session, damaging his brain.) This is clearly a passive standpoint.
However, when paired with the right Aspect, an Heir starts to exhibit active qualities. Then, an Heir is no longer the one receiving the aspect. They are now providing the aspect for OTHERS to inherit. (See: Equius providing Void unknowingly for Vriska to keep her Magic Cue Ball.) Heirs also, depending on their aspect, have the ability to control their aspect like a Witch can.
It seems that Heirs can literally become their aspect. John is able to turn himself into wind, fully becoming his aspect.
Time is literal. It has to do with timelines, the timestream, and time travel. Time players house the “Beat Mesa” equivalent on their planet, allowing a Scratch to occur if necessary. Also, as was stated by Doc Scratch himself, every successful session has a Hero of Time on their side.
An Heir of Time would be provided with Time, and then provide Time to others. When provided with Time, they would suddenly find themselves Time Traveling and creating Paradox Clones unknowingly. They would need to harness their skills to take mastery over this aspect. Once that is achieved, they would be able to provide Time for others. They can create clones of others, give other players necessary stalling time when necessary, as well as send players back in Time when needed.
And finally, an Heir of Time would have the ability to turn themselves into Time. Now, since turning into a clock is rather silly, I think they would instead take on qualities of Time itself. Patient, long lasting, flexible, as well as dominant and very powerful.