Why do you think Harrenhal was left without a lord between the end of the Dance and the Lothstons getting it in 251? From the royal family’s perspective, what could have been done with it?

Between 131 when Larys Strong was executed, and 151 when Lucas “the Pander” Lothston was made Lord of Harrenhal in exchange for marrying Falaena Stokeworth and probably claiming Jeyne as his own daughter, Harrenhal was royal property, just like Summerhall or Dragonstone or King’s Landing. 

Aegon III and his brother Viserys the Hand were probably focused on using Harrenhal’s revenues to help rebuild the royal finances – after all, while Tyland Lannister did send portions of the treasury to Casterly Rock, Oldtown, and the Iron Bank, we never find out what happened to that money and whether the crown got any of that back, and a lot got spent by both the greens and blacks during the war.

The year 131 and after was a bad time – the aftermath of the Dance, an especially hard and long winter, a Winter Fever epidemic, the political dysfunction of the regency, the Red Kraken raiding in the west, etc. When Aegon III took over in 136, his policy of “full bellies and dancing bears” would have required revenue to work, and someone as tight-fisted as Viserys would have looked to the incomes of Harrenhal as a way to make the books balance. 

By 151, there had been 20 years of recovery, and Viserys could afford to give away this royal favor to hush up his son’s indiscretions.