Since the Thalmor list Ulfric as an "asset," or to put it simply, Thalmor agent/spy, how can he ever help to hold the throne even if he is High King?

I would reread the dossier carefully. Ulfric was at one time captured by the Thalmor and 'turned' however upon being released he returned to the Empire, ran the Forsworn out of Markarth and is vehemently Anti-empire/anti Thalmor and wants to drive them out of Skyrim. The dossier shows he is no longer a reliable 'asset' and his defeat of the empire is as undesirable as a quick imperial victory. Were he a Thalmor puppet his victory in the CW would be cheered on by the Thalmor.

The Markarth incident that just so happened to be reported to the Thalmor?

His cell was likely much nicer than most. His father's death during his time inside was likely what turned him away from being "reliable" but as they said, Ulfric's victory is as unwanted as his death. The Thalmor want strife.

Certainly announcing the new High King to be a Thalmor spy, complete with the documents to prove it would cause it should Ulfric win the day.