It's a nice quote, and correct as far as it goes. "We raise these questions not in order to provide definitive answers, but in order to stimulate questioning" is an annoying trope. However, a few thoughts:

• There may be some value in finding definitive answers offputting. Namely, if one values definitive answers too highly, one may be excessively compelled to prematurely proclaim one's answers definitive! But this isn't to say that definitive answers would not be desirable when they can be achieved.
• I doubt the attitude he describes is as prevalent in philosophy departments as he suggests. The vast majority of publications in current mainstream philosophy, whatever else you may say about them, do appear to me to be concerned with the enterprise of providing actual answers to questions. (Refining the precision of the questions themselves and knocking down failed answers to them both count as aspects of this enterprise.) If philosophy isn't coming up with actual, definitive, no-longer-questionable answers very often, it may be because the questions are actually very hard, or because philosophers are bad at answering them. But celebrating ignorance in favor of rhetorically pretty question-asking is not a frequent feature of any of the philosophy I read.
• Also, Alain de Botton is an idiot who I've been wanting to gripe about for a while now. I listened to him on the Philosophy Bites podcast and he seriously seems to believe that if religious institutions are weakened any further, we won't have community or nice architecture any more. While acknowledging the factual correctness of atheism, he doesn't want people to respond to their newfound atheism by actually changing any of their behavior surrounding religious institutions and rituals.
• The questions he claims Oprah Winfrey raises are, if you click through: "how do we live with other people, how do we cope with our ambitions, how do we survive as a society". These are all fine questions, although I don't know what they would reduce down to if formulated more precisely, but it seems just silly to think they're principally philosophy questions. Serious people are working on all three of them, just not mostly in philosophy departments. De Botton seems deeply bored by generality and abstraction -- but one thing philosophers do best is figure out how to see specific problems as special cases of more abstract ones. I think he just doesn't like philosophy very much, and (in keeping with his overriding concern with keeping religious institutions active in an atheistic society) he would prefer philosophers who remind him more of life coaches, religious sages, or the like. (When he says "how do we survive as a society", I don't think he's referring to existential risk!)

The answer always comes back: 'The point of mathematics is to prove interesting things, not build things.' I can't help but think 'No. It can't be!' Imagine if you applied that question to other areas – is the purpose of civil engineering to prove interesting things about bridges?

So, sort of familiar, but not backed up by the same specifics, or even intent. Also, more passive aggressiveness.

If it is a good idea to hold off on proposing solutions, then why isn't it okay to have a division of labour between those that merely discuss a problem as thoroughly as possible (in this case, some philosophers) and those that settle on a final solution (in this case, some scientists and engineers)?

Note: I believe that philosophy has solved some problems and that these solutions are usually the fundamental principles of an immature science (at which point they stop considering such problems as being within the domain of philosophy).