> Let's just say the price is a random walk
This is where your logic is faulty. On the time scale of for example days, the price is definitely not a random walk. If stock XYZ has a price of $100 at 9:00 am, and you put in a buy order at $99.50 and a sell order at $100.50, there are a number of ways for it to go:
1. The price drops to $99, your buy at $99.50 triggers and you now are long (expect the price to rise), the price rises to $101, your sell at $100.50 triggers, you make $1.00 and are back to your original position.
2. The price rises to $101, your sell order at $100.50 triggers and you are now short, the price drops to $99.00, your buy order at $99.50 triggers, you make $1.00 and are back to your original position
3. The price drops to $99, your buy at $99.50 triggers and you now are long (expect the price to rise), the price continues to drop to $95, you now have a share of XYZ worth $95.00 which you are offering to sell at $100.50. Either you exit your position (and lose $4.50) or stay in the position (in which case you have $99.50 less available to invest than before. Whether or not the position makes sense to keep, you are now a long-term investor, not a market maker.)
4. Same as number 3 except that the price rises to $100.50, your sell goes and you are now short XYZ, the price continues to rise to $105.00 and again you either exit the position and lose money or don't exit and are long-term short XYZ (and never make the money back if the price never drops back down).
In order to make a profit, 1 and 2 have to happen 10x as often as 3 and 4 for the particular values mentioned above.
If you widen your spread, you will make more money each time the random walk of the market goes from one side of your spread to the other, but that will happen less often (whereas large one-way market moves that more-or-less permanently move the price to a level where it no longer crosses your spread stay roughly as probable as before).
If you narrow your spread, you will make money more often, but less each time. HFT firms are an extreme example of this - they have very (very!) narrow spreads that they trade on millions of times per second, pocketing a fraction of a penny each time (unless the market moves, in which case they lose money just like any other market maker, but events that happen every minute are "rare" when you are working on the timescale of microseconds).
There is a phrase, "picking up pennies in front of a steamroller", used in finance. The basic idea is that there are pennies (spreads) lying on the ground for you to pick up, but the steamroller (an actual movement of the market) could come along at any time and crush your profits.