When Blackbeard makes his move, he makes MOVES.
Also, look at how interested/confused he is by the Seraphim.
Laffitte's hypnosis, if it works on Koby, is huge for Blackbeard, because he can learn all about the Seraphim and Sword from Koby.
If Blackbeard knows what Vegapunk is working on, I highly doubt he'll just ignore it, it's a gigantic threat to him. The Seraphim are designed to replace the Warlords and stand against the Yonkou. If Blackbeard gains the Seraphim's power instead, he completely flips the entire table. Obviously, he can't get Gorosei clearance (good luck kidnapping a Gorosei), but...
All Blackbeard needs to control them is Vegapunk. Then, unless he's fighting the Gorosei, Blackbeard doesn't need to worry about them turning on him. Once again, Laffitte can either hypnotize Vegapunk, or, if that fails, Blackbeard can use threats.
Fits well narratively, too: Luffy makes Warlords his allies through fighting them, Blackbeard uses dirty means to control clones of the Warlords.
Blackbeard kidnapping the Seraphim was foreshadowed by his attempt to steal Boa's power: instead, he'll steal her clone.
This is all the other shit I think his crew is up to, by the way:
-Moria and friends attack Sphinx, resurrect Ace and Whitebeard as zombies.
-Small force to explore Elbaf that clashes with Kid
-Catarina Devon is in Wano, and has replaced Hiyori as princess using her Fruit. Catarina declares Tama to be a Kurozumi, and Yamato to be Kaido's child, saying her brother, Momo, has allied with the Beast Pirates and the Kurozumi Clan over the Kozuki Clan and Wano. This allows the Blackbeard Pirates to begin taking over the country, and is a throwback to Bon Kurei impersonating Cobra in order to trigger the massive war in Alabasta, since they have similar Fruits.
I have so much shit riding on Blackbeard.