Well, I think it's pretty simple really. It wasn't a part of the plan.
Blackbeard is hunting seemingly all the most powerful Devil Fruits. But, he's not planning on keeping all of them. Remember, he was thinking about selling Law's Op-Op Fruit. One of the most powerful Fruits in the world, and he wasn't even sure if he wanted to keep it or not. And while he did want Ace's Flame Fruit, he pivoted and adapted pretty quickly once Sabo got it. He didn't go after Sabo after he got the Flame Fruit, Blackbeard just moved on.
To me, this says that Blackbeard is hunting for very specific Devil Fruits. It's probable that every Fruit that he's given to his Titanic Captains has a specific function that he's planning on using down the road for his master plan. Maybe that's giving him too much credit, but we'll see.
So, it's likely that Blackbeard doesn't really care about the Phoenix Fruit. Despite it being powerful, it's not a necessary Fruit for his master plan, so it's a secondary target, at best. He'd probably take it, if it was a easy get, but if it's too much work, it's not worth it to him. Blackbeard went all out against Whitebeard himself, most likely because the Quake Fruit is a key part of his plan. The rest? Good if he can get them, but not the priority.
So, if the Whitebeard Pirates don't have any Fruits that he's actually after for his plan...why risk more of a fight? Why pick a fight with a strong opponent, when you don't have to? When it's easier to just run away? Blackbeard is ALL about the profit. He's all about the goal. And he's also a little bit cowardly at time. Probably from being sensitive to pain. He'll take risks, but only when the risk is worth the reward to him. At this point, hunting down the last vestiges of the Whitebeard Pirates isn't worth it to him. Too much risk, too little reward. Their spirit is already broken twice over, once when their "father" died, and again when Blackbeard beat them during the Payback War. They're done. They're irrelevant, quite frankly. I feel like Blackbeard believes they aren't even capable of trying again...so why bother with them?
So, yeah. They probably survived by a combination of the few remaining fighters retreating. And Blackbeard not bothering to pick off most of the stragglers, preferring to just let the fight end. Blackbeard's crew didn't get wounded as badly as the Whitebeard Pirates did, but they were probably still tired after a fight like that. Better to let them run, rather than chase after them and start to lose members of his own crew because of it.
