"Intuitively, everything else being equal, if a stock has higher drift, shouldn't it have higher probability of finishing in-the-money (and higher probability of having higher payoff), and the call option should be worth more?"

All these other answers are focusing on the wrong aspect of the question - it is true that the maths makes the drift drop out from the BS PDE, but that doesn't explain why intuitively that feels wrong.

The main reason for this is that you are confusing the investment thought process with the pricing thought process.

To understand it better, look at it this way: imagine you have two stocks $S_1$ and $S_2$, with different underlying Brownian motions, same sigma, but different drifts $\mu_1$ and $\mu_2$ (say $\mu_1 > \mu_2$). This does not introduce arbitrage, as the underlying Brownian motions are different. Since everything but the drifts is the same, the BS PDE ends up being the same, so

1. the call options on $S_1$ and $S_2$ have the same price, which is the same as saying the drift has no impact on the price. Your intuition, though, tells you that:
2. the call on $S_1$ should be worth more because there's a higher chance of it expiring in the money (equivalent to saying the drift matters).

Both 1 and 2 are correct, and there is no contradiction between the two. 1. is given by the usual BS logic, where the price is set such that there is no arbitrage and the derivative is perfectly hedged. 2. is given by the process followed in the investment process. The option price does not care about what the trade off between risk and reward is in the market, it is oblivious to any market prices of risk. Since the market price of risk is directly tied to the drift, this should make it clearer why drift does not matter in this world.

"Intuitively, everything else being equal, if a stock has higher drift, shouldn't it have higher probability of finishing in-the-money (and higher probability of having higher payoff), and the call option should be worth more?"

All these other answers are focusing on the wrong aspect of the question - it is true that the maths makes the drift drop out from the BS PDE, but that doesn't explain why intuitively that feels wrong.

The main reason for this is that you are confusing the investment thought process with the pricing thought process.

To understand it better, look at it this way: imagine you have two stocks $S_1$ and $S_2$, with different underlying Brownian motions, same sigma, but different drifts $\mu_1$ and $\mu_2$ (say $\mu_1 > \mu_2$). This does not introduce arbitrage, as the underlying Brownian motions are different. Since everything but the drifts is the same, the BS PDE ends up being the same, so

1. the call options on $S_1$ and $S_2$ have the same price, which is the same as saying the drift has no impact on the price.

Your intuition, though, tells you that:

2. the call on $S_1$ should be worth more because there's a higher chance of it expiring in the money (equivalent to saying the drift matters).

Both 1 and 2 are correct, and there is no contradiction between the two. 1. is given by the usual BS logic, where the price is set such that there is no arbitrage and the derivative is perfectly hedged. 2. is given by the process followed in the investment process where what matters is the trade off between risk and reward (or the market price of risk). Because of hedging, the option price does not care about what the trade off between risk and reward is in the market, it is oblivious to any market prices of risk. Since the market price of risk is directly tied to the drift, this should make it clearer why drift does not matter in this world.