Is the tongue important? Ha, it depends on who you ask! When people ask me, I always say yes, of course it is! But, like with anything on the trumpet, when you think about it too much, it hinders your playing. Everything, and I mean everything, has it's balance point. The same is true with the tongue.
The tongue is an integral part of your trumpet playing anatomy. It does much more than we think it does. Many experiments have been done regarding tongue movement and they all say the same thing - the tongue is important. But..... it also depends on the player how much of a role the tongue will play. Meaning, some players need to use more movement of the tongue (tongue arch) than others. This may be due, in part, to the shape of the oral cavity, the size and length of the tongue, the length of the frenulum (the part of the tongue attached to the front bottom of the mouth), and size of the teeth, etc. I always explain and demonstrate to my students what I need to do, but also tell them that they need to find their own balance (through the way it sounds). This is why I encourage them to take a lesson with someone else and attend summer clinics and festivals, to figure out what other people do and how that relates to them (not just in regards to the tongue!).
Ultimately, from my point of view, the sound is the overall equalizer. The tongue may in fact act as a "fine tuning rod", but you must alway keep your focus on the main goal, and that is the sound. If your concept is clear and you can hear what you want to produce, the body has a way of making that happen. At this point, we just need to stay out of the way, both physically and mentally.
We trumpet players tend to overthink things too much. Is the tongue important? Well, of course it is...