First, some data:
http://sociology.ucsc.edu/whorulesamerica/power/wealth.html
http://www.good.is/post/americans-are-horribly-misinformed-a...
In a nutshell: the top 20% of the populace controls 85% of the US's wealth.
1. Increasing wealth concentration = increasing concentration of political power => democracy undermined. US is excellent example - the populace can vote, but only for candidates preselected by funders in the top 20%; populace can advocate, but top 20% has much better access, influence (due to both campaign funding and 'revolving door' jobs when leaving office), and in some cases actually writes the legislation (bank bailout being a major example).
When that happens, you get capture of the government and regulators, financial excess & crisis, and symptoms of banana republic:
http://www.theatlantic.com/magazine/archive/2009/05/the-quie...
2. 'As long as each generation is relatively wealthier than the prior one, I don't care about the spread within that generation.'
That's held true for decades, but is not guaranteed to continue forever. Problem of induction: no amount of confirming observations can prove a theory true, while one single refuting observation can disprove it. Eg, even if all strata of every generation has always been relatively wealthier than the previous generation, that is no guarantee the same thing will continue indefinitely in the future.
Highly concentrated wealth also skews some economic statistics, masking the plight of the lower and middle class. The latter can even be in decline while the top 20% are making out like bandits by outsourcing chunks of our wealth-creating 'making things' industries to higher-profit-margin manufacturing centers, and that wouldn't show up on some stats based on averages.
The workers get laid off while the CEOs, boards, 'pay consultants', and shareholders pocket the increased profits. And when the top 20% controls 85% of the wealth, guess who those shareholders are. Bottom 80% is increasingly locked out of the wealth that can be generated from ownership, and forced to subsist as wage slaves.
There are arguments that is what is beginning to happen now.
3. 'This continual improvement in wealth amongst my own family is proof enough that the very rich aren't impinging on what really matters, which is improving living standards for the majority of people.'
Your standard for 'proof' is pretty low. That's a single data point anecdote. C'mon now.
4. Extremely high wealth concentration has historically been associated with revolutions and other unrest. The French Revolution comes to mind. I'll let the amateur historians here argue over causality, but it's really not a road we want to go down.
For anyone who believes in the adaptive, corrective power of free markets and democratic government, increasingly extreme wealth concentration should be a concern, since it can and does undermine exactly that.
PS - I'm not implying wealth redistribution is the solution, since that's just a poor kluge that addresses the symptoms rather than the underlying problem/s. But I do think the reasons for this accelerated wealth concentration need to be clearly understood, and in some cases neutralized - unpunished financial fraud that led to the crisis, CEO pay based on board & pay consultant connections, free trade with countries that allow effective slave labor, among others.