I dunno, I think most people among the t-shirt's target audience dig more "subtle" and "clever" humor, but maybe it's just me. Try it out and see, that's the only way you'll know :)
With all due respect to Snapchat's users and their puppies - they're irrelevant. Successful businesses function in one of two ways: they either create a solution to a problem that bothers certain parties and sell that solution to them, or they create a problem and then "invent" a solution. Any real business on this planet is about supplying something that you know there's demand for. If you know for sure that you want to be a big-problem-solver, then go for it, you'll find your market as you go. Also, this might serve as an inspiration: http://www.technologyreview.com/magazine/2012/11/
Have you tried Twitter Bootstrap? It follows the "mobile first" philosophy that you're after, and it's also quite easy to disable responsiveness entirely, if needed. I'm mainly a back-end developer and I (passionately) hate doing front-end work, yet with Bootstrap I feel like it is taking a huge load off my shoulders. AFAIK it does a great job at supporting the various browsers, widths, orientation and so on, also comes with some boilerplate JS for stuff like highlighting a mandatory form field, handling modals, buttons etc...