Peace Aaron,
Understanding Salat as 'merely' a commitment is kinda vague and short-circuited. I believe commitment/obligation IS the most difficult part of ones life on a daily basis until death. Commitments/obligations starts the moment one wakes up and ends once asleep, and so on. It's a BIG responsibility at every second of ones Way of Life.
If you can't find water (rather unusual though in this generation) to cleanse yourself, what else would you do? Most now-a-days would rather use only paper towels, but dry sand/mud/dust seem to work better to remove stickiness, sweat, etc. (Tayyammum). Try it out, I already have. A plain simple method of cleansing that has nothing to do with ritual prayers, oration, linkages, etc. At least that's the way I understand it wholly.
On the same lines, if one had no toothbrush/toothpaste, how would one "brush" his/her teeth if need arise? Common sense, right.
