But I have seen this kind of thinking used as a crutch or an oversimplification in training. Punches are generally meant to be thrown fast. You want to use the proper technique, of course, but part of the technique IS being fast. Not every punch needs to be but most do. It's like a pitch in baseball. It is meant to go generally fast. There are off-speed pitches for sure but the goal is to throw the ball fast enough so it's difficult for the batters to see and hit.
Throwing fast punches isn't just trying really, really hard to go fast, there is technique to it. You need to learn how to relax certain muscles completely while activating others quickly. You have to develop those fast-twitch muscle fibers and neural pathways. There is a whip timing effect that comes from snapping your hand back after the punch lands. This is all technique.
The same is often true for power in boxing as well. Punches are thrown with the intention to hurt your opponent. Boxing experts always like to tell you that not every punch has to be powerful and again, there is certainly truth to that statement but if the majority of strikes do not at least bother your opponent, they will walk right through you. Even if you are using throwaway punches to try and set up a power punch, you likely won't be able to get anyone to react to your set-up without giving them something they don't want to get hit with.
You also don't generate effective power by muscling it and just trying to punch hard. Throwing a punch properly is what gets you power, using effectively-coordinated body mechanics. Things like specific core rotation and leg drive. Breathing properly is very important too. It all must be learned and practiced. Some people find this more natural than others but that doesn't mean that there is no technique in their power. Are there fighters with a lot of power but generally sloppy technique? Yes. Sometimes this can compensate, sometimes it can't but in any case, the power puncher with sloppy technique would always be able to generate more power with better technique.
There is too often an all-or-nothing thinking when it comes to both speed and power. That throwing fast punches or really powerful punches must come at the expense of technique. This can be true but it is not inherently true, nor are these things inversely proportional. Just because you train to throw powerful punches or fast punches, doesn't mean that you are wild. Nor does it mean that you don't care as much about form and technique. You can throw explosive strikes with technique and once again, technique is part of what makes them explosive.
It could be said that you can throw faster punches with a shortened range-of-motion and harder punches with a little more range-of-motion than normal. Again, this is part of the technique and does not equal wild and out of control.
Going back to the beginning, does it make sense to learn the form slowly first before trying to go fast and hard? It certainly does and this is logical but it is an unrealistically oversimplified understanding of how one learns something in boxing. It assumes you can get the perfect technique down, then try to go faster/harder. But learning is not binary. I can teach someone how to throw a cross in slow motion for months and it is still not going to be perfect. And while we are training many of the muscles and mechanics, we are still not doing anything for the fast-twitch muscle fibers and neural pathways required to throw it at game speed. A slow motion cross isn't a cross at all. It is a body movement. What makes it a cross is that explosive burst of acceleration times mass, which is as much part of the technique as rotating your shoulders is.
That being said, if you are new to boxing and still unsure about the basic mechanics, absolutely slow things down at first. I have also seem some new folks, usually guys, who seem to go into Beast Mode when trying to throw a punch and are off-balance, wild and often sloppy. This is definitely the kind of person I would tell to slow down for a while and to avoid getting into a state where you are just trying to explode without thinking. Just make sure you learn how to add the explosiveness properly after slowing things down.