The question asks which is more effective: Doing MMA or doing multiple different martial arts.
There are a couple of different interpretations about what is meant by "effective" in this context, however. First, it can refer to how well all the different styles of martial arts are integrated into a cohesive system whereby all the techniques work together and can easily and quickly be switched from one to the other. Alternatively, it can refer to how good it is for either sport or self-defense.
As for going out and learning multiple styles and integrating them into a cohesive whole, the problem you're going to encounter with that approach is that all the individual, "pure" martial arts don't combine and work together very well without doing a lot of adaptation.
For example, if you want to combine western boxing with judo, you're going to find out through trial and error that the boxing stance is all wrong for an MMA or street scenario. You're too upright in a pure boxing stance, and that means you're more easily taken down by someone who shoots in on you. It also means you'll have to take more time if you want to shoot in on your opponent, telegraphing your moves as you crouch down and lean forward to shoot. So you'll want to modify your boxing stance by sinking down a bit in your legs, spreading your feet out some more, and leaning forward slightly so that you can quickly sprawl against your opponent if they decide to shoot in on you. In boxing, shooting in on someone is against the rules. They don't have to worry about that happening. And so they have optimized their stance so that they can have an advantage in a pure boxing match. For real-life or MMA, it doesn't work well. You'll have to modify it.
Form dictates function, and vice versa. The rules of each martial art, and the assumptions about what those arts are going to be used for, cause those martial arts to look the way they do. If you alter the rules to allow for a wider range of techniques, those martial arts may have to change more or less everything about what they do and how they do it.
This approach is very interesting and cerebral. It will take a very long time, lots of trial and effort, and lots of analysis and research. But ultimately, you'll probably end up with something that looks basically the same as MMA. So, why not just do MMA instead? The people who teach MMA have already done all of this analysis for you. And they didn't do it alone. There are thousands of others that came before them. The most time efficient and practical thing is to just go directly into MMA instead of trying to reinvent it yourself.
But there is a little twist here. Self-defense and MMA do not have the same rules, goals, and strategies. Training only in MMA (from sport-based MMA schools) will probably give you bad habits that will be dangerous in self-defense situations. If self-defense is your primary goal, I recommend learning Gracie Jiujitsu first and/or seeking out an MMA school whose primary interest is in self-defense. They're much more rare, but they are out there. You just have to ask around.
Gracie Jiujitsu is primarily focused on self-defense. They don't optimize for sport until much later on, after blue belt. Their foundational skills that they build at the white belt level allow them to recognize later on when a technique is dangerous for street fighting. And Gracie Jiujitsu is a kind of MMA. They begin at white belt with defense against punches, kicks, and so on. But later on, they actually teach you how to punch and kick. They've integrated it into their style nicely, and it continues to evolve today. They even have small circle joint manipulation and self-defense drills like you see in classical Japanese jujitsu.
Then later on, you can decide to add MMA to your Gracie Jiujitsu training. Or you can go and do MMA entirely. It's up to you. The stuff you learn in Gracie Jiujitsu will make you better in MMA, even if you only care about sport and not self-defense. And that's how most people approach it nowadays.
Many people take Muay Thai classes, also. But in my opinion, you'll get most of what you need of Muay Thai by taking MMA classes instead. Same with western boxing. But Jiujitsu is a different thing. Jiujitsu is much more technical and intricate, which is why I recommend taking it all by itself.
Hope that helps.
