Is there absolute utilitty maximisation in portfolio diversification or is that just a risk control mechanism?

It's purely for risk control, but most people are extremely loss averse and so do well to diversify.

Could I pick one random stock and put a whole lot of money in it? I suspect I may be commiting the law of large >numbers here (or the gambler's fallacy).

You could. It's a bet with positive expectation and a really risky one. But people do much dumber things with their money. Having said that, I'd recommend an index fund instead if you're plopping a whole lot of money in.