Pardon the long reply, but I have a few comments on some of the OPs statements.
A. The OP is right that humans "have the tendency to try and package everything into neat little boxes" - In a sense, that's exactly what music is. One of the more popular theories about music is that it is ways of organizing sounds. From that, a popular theory for rhythm in music is that it "describes the pattern of events in time." (Handbook of Acoustic Ecology 1999). This leads to yet another theory that there is a wide range of rhythmic types, from "metric" (which I think the OP refers to as "mathematical") to "non-metric" (which I think the OP considers "creative"). So given these theories, some musics can be observed as organized to represent the "organic" rhythms of "nature". Other musics are organized to mimic the precision of machines. Still other musics are not representative of anything physical, and are instead abstract or conceptual. But whatever the objective, music is an attempt to organize sounds, whether melodically, harmonically, or rhythmically. The packaging of the box may not always be neat, but it is still a box.
B. "Is music becoming too mathematical?" - Using the concept of "metric" rhythm, which "involves the subdivision of the measure into equal units marked by a pulse called the beat" (Handbook of Acoustic Ecology again) I can safely state the following.
1) There are types of music that value metric rhythmic consistency and precision. - The history of Western music is littered with tendencies towards the achievement of rhythmic precision. In some cases, it has reached a point of obsession. But its not a question of good or bad, but whether the specific music work is best served by it.
2) There is an ever-growing demand for this kind of metric approach to rhythm in contemporary western popular music - For example, dancers may demand constant tempo to facilitate uninterrupted movement. Pop mega-production concerts thrive on electronically generated time code to synchronize musicians with light shows, videos, pyrotechnics, and stage movements.
3) The facilities to achieve metric thythmic consistency and accuracy have become increasingly available - For example, the metronome provided an accurate guide for musicians to achieve rhythmic consistency. As mentioned above, synchronization was become increasingly accurate to milliseconds. More recently, quantization in music computer software has enabled human musical performances to be modified to achieve inhumanly consistent rhythmic accuracy.
So my answer would be that while metric rhythm is hegemonic in conteporary Western music, it is still not the only existing rhythmic approach in the world.
C. "Do you really think a tribal drummer in New Guinea sits down with a special drum book and a metronome while counting "1 e and a, 2 e and a..." praciticing his paradiddles and rudiments? Or does he tap into a deeper state of energy that we seem to have forgotten about and play with every single atom of his body?" - Chances are if that tribal drummer lives with a tribe, he/she probably learned drumming from elders, and follows a long history of musical traditions. While these traditions may have been passed on by rote and not by notation, they nevertheless feature various methods of musical organization, which include indigenous rudiments and rhythmic patterns. Among a number of indigenous tribes of my country (Philippines), there are virtuosos who have devoted much of their life to the practice and teaching of the discipline of their instrument. So even ancient traditional music may emerge from musical convention and discipline.
D. Finally, I'm not trying to bellitle the OP, but a statement like "I use drumming as my creative outlet and always try to play without any preconceptions or limitations for myself. This is one reason that I never felt the need to learn any sort of music theory" is begging for a lesson in theory. All the points I raised above are based on theories. That theories provide possible answers to the OPs questions should encourage more people to study them. Theory is not the antethesis of creativity. And to eschew them in an attempt to avoid conformity denies the wisdom of the past. The challenge of any creative musician is to build on, apply, and critique the foundations of past knowledge in an attempt to forge new directions that can be applied in the future.