I can only speak for myself, but no, I'm not worried about it. Bass-less bands have existed for years, The Doors being one of the most notable acts from years past. Sleater-Kinney has had a very long and noteworthy career and never had a bass player - The Jon Spencer Blues explosion too. I still think Local H's As Good As Dead is a classic album from the 90s.
Many bands make line-up choices based upon both artistic and practical reasons. A lot of these bands formed in the way they did because they couldn't pick up the required players back when they were just just jamming in the basement of their local towns. Once they started getting success, they just decided not to mess with the formula. The White Stripes have always been sort of a modernized, electrified version of delta blues with some alternative rock thrown on top. Throwing a bass player into the situation may mess up the vibe they were going for. Now, White usually has a bass player in his solo band, which I do personally prefer.