Supplementing System Down's answer:
Let's discuss example of Volantis (which you have cited as well), which remains the closest in culture and government to the Freehold of Valyria.
While the Free cities did fealty to the freehold, they were essentially free. They had their own economy and governments and even went to war independently. For example the Rhoynish wars in which the colonies went to war against Rhoynar for trading supremacy in Rhoyne region.
The last battle of this series was when Volantis went to war against Rhoynar city state Sarhoy, called the second spice war. Rhoynish Prince Garin lead his men against Volantis and promptly defeated them and their three dragons. Volantenese ran to their black walls and called for help from Valyria. Mother Valyria came to rescue of her daughter and three hundred dragonriders changed the tide. This shows while there were strong relations between Valyria and her daughters, daughters were independent in their own ways.
After doom of Valyria, began the reign called "Century of Blood". Volantis considered herself to be heir to legacy of Valyria and went to war to conquer all Valyrian territories.
They conquered Lys and Myr into their "New Valyria". They also sent expeditions to recover lost Isles of Valyria.
Afterwards they marched to next target, Tyrosh. That's when Pentos realized that if Tyrosh fell, they would be next. So they joined Tyrosh. While Volantis was busy holding off Tyrosh and Pentos, Lys and Myr rose up in rebellion to win their freedom.
Braavos, Qohor, Norvos, Storm King Argillac the arrogant and Young Lord Aegon Targaryen on his dragon Balerion also joined the fray against Volantis as creation of a New Valyria threatened all of them and their interests. Only Lorath remained neutral, mostly because they were on the Farthest edge of the conflict and have historically remained isolated. Following were their reasons I suppose:
Sealord feared that a new Valyria would try to enslave Braavos because Braavosi were descended from run-away slaves of the Valyrian Freehold.
The Nine Free Cities are the daughters of Valyria that was, but
Braavos is the bastard child who ran away from home.
Qohor and Norvos were naturally concerned about their freedom and unwilling to accept Volantenese supremacy.
Aegon Targaryen was concerned about his own holdings as Dragonstone was also a part of old Valyria. Also on account of being one of the old blood and Dragonriders, he probably considered the claims of Volantis to be heir of Valyria as ridiculous. But since Aegon never tried to rebuild Valyria and instead turned to forge his own new realm, we can say he never had any such ambition.
Storm King felt threatened as in case of a successful reunification, Volantis would be a direct threat to his Kingdom, right across the narrow sea. Who was there to ensure that New Valyrian conquests won't spill across the narrow sea, well beyond the original boundaries of the freehold?
Lorath did not feel inclined to join the fight because they were an Island nation, cut off from most of the continent Essos. Not to mention they always maintained low profile in regional affairs. However, ordinary Lorathi citizens did fight on side of Braavosi or Norvosi despite neutrality of the government.
End was like WW1, peace favoring Volantenese faction Elephants wrested the power away from War-loving tigers, and ended their ambition to reunite Valyrian territories.
Later Kingdom of three daughters was formed with union of Myr, Lys and Tyrosh. It wasn't aimed at Valyrian reformation but instead on countering Volantene threat. However, it too expanded by conquest and soon after, a rival alliance between Pentos, Braavos and even Lorath ended them for good. That shows us Free cities won't let any of them become too powerful to overwhelm the rest as they have no desire to unite under one realm.
As you can see, formation of a new Valyrian Empire threatens everyone concerned, causing serious issues for people harboring such ambitions which are even more pronounced if they don't have dragons.
With the Political impacts aside, why should the Free cities wish to unite?
- All Free Cities have different cultures.
- They all have different languages.
- They all have different religions.
- They all have different forms of administration.
- None of them would be willing to accept suzerainty of the other.
That would be like suggesting IRL that Belgium, Holland, Germany, Austria, Switzerland etc. should unite today because they used to be together in Holy Roman Empire centuries ago. Or that Newzealand, Canada & Australia should reunite with UK as they were once colonized by British people.