Inequality of opportunity - I'm all for that fight. Inequality of outcome? - No, I won't ever support that, UNLESS one's gains are ill-gotten.
But one cannot have, as you say, a society where the aim is for "everybody to become rich". That idea doesn't even exist in the realm of Utopia. Wealth will always be distributed normally (or some variation of the normal distribution curve). A lot of people just don't have the intellect, work ethic or character to create wealth, no matter what we do in the fight against inequality.
Furthermore, wealth is relative. The poor of yesteryear would have considered the poor of today as living like kings. Only a part of that is due to the welfare state. Most is due to mankind's technological progress - aided by market ecomomies operating under supportive institutional structures - over the last couple of generations or so.
