Daenerys brought the dragons back through the blood of the first men and three sacrifices to the old gods
A long before Daenerys was born. A short albino woods witch who was known as the ghost of highheart made the prophecy about the prince that was promised. One specific part she mentions is that the prince that was promised will be born from the blood of aerys and rhaella targaryen. Prince Jahaerys, their father makes the decision to have them marry each other.
While it may look like the ghost of high heart suggested the marriage through blood purity. I think it was more align towards to strengthen the blood of the first men. High heart is located in the riverlands and is tall hill that was sacred to the children. The ghost of high heart is rumoured to be a children of the forest, however it's unlikely if that's true.
The prophecy she made may have likely came from the old gods of the forest themselves meaning that the ghost of high heart is more likely a green seer. Which is why her prophecying that the prince that was promised will be born through Aerys and Rhaella, it wasn't through because blood purity but rather through the blood of the first men.
Aerys and Rhaella are trapped in an abusive marriage. While this marriage does pay off in the end through the birth of Daenerys Targaryen who is definitely the prince that was promised it ends up being the one to bring the dragons back. It also the tragedy and how prophecies have fuelled targaryen madness.
When Daenerys eventually does end up bringing the dragons. It had nothing to do with her blood of old valayria but rather through the blood of the first men and the old gods. In case you don't know, Daenerys actually has first men blood in her through her great-grandmother Queen Betha Blackwood.
I found it interesting that Prince Aegon from the dunk and egg stories chose to marry a highborn lady from a minor house like house blackwood. However, there ended up being a reason and it was through what house blackwood represents through their religion. They worshipped the old gods and are also the first men descdendant.
Daenerys being a first men descendant is what actually woke up the dragons. Daenerys choosing to sacrifice Khal Drogo, Mirri Maz Duur is similar to when there was a time where the first men would blood sacrifice as a way to worshipped. However the blood sacrifice came from criminals like thieves, rapists, murderers etc.
The three people who Dany choose to sacrifice actually did crimes which would be considered punishable by death through the old gods. Khal Drogo invaded villlagers through rape, enslaved and murder. Mirri Maz Duur made false promises which led the death of Daenerys's unborn child. Than you got Daenerys who volunteered to be the last sacrifice. Her crime is letting the other crimes to happened under her watch. Not to mentioned kinslaying.
Daenerys is unknownly using blood sacrifice to worshipped the old gods which leads to the dragons waking up all these years ago. Daenerys' beliefs and views like her freeing the slaves, her not treating her dragons as weapons but rather through sacred bonds aligns more towards the old's belief. They considers crimes like slavery, incest and kinslaying as abomination and punishable by death.
The return of the dragons wasn't through old valayrian but rather through the old magic of the old gods.