I’ve had some training in jiu jitsu,mui tai, and have a little wrestling expieriece.
Those (presuming you mean Muay Thai) are the top three things to train in. The only ones I would add to the least of consideration are Judo and boxing, meaning if the BJJ gym sucked and there was a good Judo gym I would go there. If the MT gym sucked and there was a good boxing program I would go there (some people out rightly prefer boxing to MT I am just the opposite).
Psyproffesor's post has some solid points in it. I would add in addition to cream puff training the rules of a style's competition are telling. For example, I have never met a TKD guy who kept his hands up. I could go on with examples of other arts and the holes they leave, largely because of the rules of competition, but I wont.
As to Katas, they are more like shadow boxing than the heavy bag. The idea behind them is the same, you are ingraining a move and committing it to muscle memory. There is one giant and very important difference. Boxing/MT foot work, punches, etc that is being ingrained work. Horse stance punching from the hip BS does not. One is ingraining impractical stuff and bad habits. Another difference is why they are being done today. In many places katas are done for their own sake, ie you see kata competitions. Can you even imagine a shawdow boxing competition or a heavy bag workout competition? The idea is absurd. It is just as absurd for Katas but people do it.
There is something to the idea that the individual matters more than the style, ie someone who simply is not tough can't take a punch, is out of shape, will probably get mauled by a tougher physically superior opponent irrespective of their professed styles. I have been involved in wrestling, Muay Thain and BJJ for over a decade and I do believe style counts for a lot. It is a matter of both what those styles teach and how they train. You wont meet someone who is a decent wrestler who is a wuss. Those guys quit. Same for boxing and Muay Thai (if we are talking about people who actual fight). People who can't hack it get weeded out. I have met many tradition martial arts people that thought they were genuine bad @ss dudes and were exposed for the paper tigers they were the moment they got popped hard in the face. It is amazing to me that in their years of training they had never been hit hard before. Things like that matter, a lot!
If you seriously want to learn to fight, find a good muay thai gym and a good BJJ gym (these days they might well be in the same building together).
MT will give you conditioning, striking, clinching, and teach you how to get hit without falling apart. It will give you sparring and some gut check moments. In Muay thai you will experince extreme fatigue and pain, you'll get beat up a bit. As a result you'll learn to fight through pain and learn the importance of conditioning. Boxing could do all the same thing save the clinch, which I love.
BJJ is essential to knowing how to fight. I wouldn't opt to go to the ground in a street fight but you might well find yourself there for one reason or another. With a relatively limited skill set it is easy to destroy someone on the ground who doesn't know anything. The biggest problems I see in BJJ training (at many but not all gyms) is three things: 1 a lack of hard training. I see a lot of pot bellys with purple belts around them. The pace is not pushed, guys take breathers the instant they start to feel tired etc. This is a matter of how some people train and not a knock on BJJ itself. One can run a BJJ workout like a wrestling practice (which is shear hell) just as easily. 2. There is often little emphasis placed on wrestling. This is a problem because all those ground skills mean nothing if you can't get the person to the ground. I have trained with a lot of guys who are amazing on the ground but had almost no ability to get the fight there. 3. There often is no striking and thus no ability to learn to take shots.
I believe those major weakness are dealt with by cross training in a practical striking art such as Muay Thai o boxing.
As someone else said train to compete. Fight a few MT fights do some grappling tournaments, eventually fight an MMA fight if you like. The comepetition will go a long ways in getting you to where you want to be.
As one last aside I personally believe in compartmentalizing training to begin with and then after a sufficient degree of proficiency has been reached blending them. This means train and learn muay thai, train and learn BJJ, then start working them together in MMA type training. I could go one about why I feel this way but I leave it at that.
No training is just the same as a street fight, but certain types are much better for preparing one for that type of scenario than others.