I actually think Contracts and Blood Money, being a pair of games, has the best balance of the two views. It had an overall story that is slowly fleshed out, but it happens in the background, and every mission doesn’t necessarily play into it. In every other game, the main enemy is actually the client for each hit, or the main bad guy is who 47’s hits are slowly taking down.
But with this pair of games, each mission doesn’t necessarily play into the story. In Contracts, the story is restricted solely to the cutscenes and the final mission.
In Blood Money, we don’t even know how involved in the conspiracy the missions are, or who the client even is. I’m pretty sure the first two have nothing to do with the Franchise or their cloning plot, and the third only relates to it because Fournier was there to ambush 47; cue the events of Contracts. After that, it gets tricky.
The rehab center and Sinistra missions seems to be only loosely tied to the conspiracy, with the talk of albinos and Cayne’s (dubious) claim that Sinistra had info about Ort-Meyer. In both cases, the Franchise can’t be the client because they are rivals with the Agency, and whoever the actual client was, 47’s going after those targets seems to be coincidental to however the Franchise was involved, unless the Franchise was just trying to take out the targets first to show that they were better than the Agency and take their business.
The next two had involvement in the plot, but we don’t yet know the details, as 47 is just whacking some of Cayne’s hitters and then one is waiting for 47 at the Christmas party. Both involve the upcoming elections, but the Christmas one doesn’t seem to have that as the reason why 47 is involved. The two Mississippi missions seem to have no involvement with the conspiracy whatsoever, and the casino mission only relates in how it shows genetic samples used for cloning are exchanging hands all over the world. The shark club mission only had the Franchise involved in that they are lying in wait for 47, but I don’t think they’re the client, making the whole mission a trap to begin with; I think they just took advantage of the opportunity. And of course, the White House mission is all about the conspiracy.
So, there are missions that relate to it, ones that don’t, and some that only relate to it in passing. You can play through it and the plot doesn’t really matter until the end, when it all comes together and makes sense, with Contracts having that same story but none of the missions relating to it until the last. This is the way to go, I think.