Isn't MC>P a better aim? Given the revenue you earn from each unit is more than cost in producing each unit?
Your intuition is correct (assuming that you mean $MC < P$ rather than $MC > P$ as you wrote). For a given level of production, profits are indeed higher when $MC < P$ than when $MC = P$, as you'd expect from an increase in the price holding everything else constant.
So why do we say that a firm in a competitive market maximizes profit by setting $P = MC$? Because we're not looking at an increase in the price holding everything else constant, we're looking at a change in the quantity, holding price constant.
It may help to think first about the traditional model of monopoly, where a monopolist will maximize profit by choosing a quantity $Q$ that sets $MC = MR$, i.e. marginal cost = marginal revenue. Then, because they're a monopolist, they can sell that quantity $Q$ at a price $P > MR$ (and so we get $P > MR = MC$ or $P > MC$). This is because the monopolist has some control over their price: they can choose to go high price/low quantity or low price/high quantity. In the process of selling an additional unit, their revenue goes up ($MR$) by less than the price ($MR < P$), since they're moving towards the "low price/high quantity" end of the scale - they get paid the price $P$ by selling an additional unit, but they had to lower their price too and so make less on the other units they sell, making $MR < P$.
That takes us back to the competitive market. In the competitive market, each firm has no control over the price and so there's no price/quantity tradeoff - whatever quantity they pick, the price stays the same. So, in a competitive market, $P = MR$ by definition. So when they maximize their profit by choosing a quantity that sets $MR = MC$, since $P = MR$ we get $P = MR = MC$ or $P = MC$.
Intuitively, the reason the firm in the competitive market doesn't shoot for $P > MC$ is that the price is real darn stubborn. You can't coax it to go higher by picking a lower quantity (like you could if you were a monopoly). So it's not "I'd like to pick a price higher than my $MC$" because you can't pick a price. Instead, it's "I may as well continue producing more and more until my $MC$ meets the price". So, for a firm in a competitive market, profit is maximized at $P = MC$.