Sure the unconditional expectation doesn't change, but that's kinda useless because it's the expectation given that you know nothing. The interesting part is studying what is next given what I know right now i.e conditional expectations. And the martingale assumption i.e. "my best guest for tomorrow is the same as right now" is honestly a pretty sensible assumption for many things.
If I tell you $TSLA is at 200 right now, it's not unreasonable to assume it will be around 200 tomorrow.
If it's raining right now, it doesn't seem too far fetched to guess it will probably be raining in 1 minute.
etc.
And because you can prove so many things on martingales, it is often very very useful and powerful when you have something that isn't quite a martingale to think of a way to make it a martingale, prove whatever and then go back to the original object.
That's a bit of an unfair example, though. If the Tesla stock is at 200 right now, the martingale property implies that I should expect it to be at 200 not just next minute or tomorrow, but also next week, two years from now, next decade, and so on. A martingale is not restricted in its time scale.
(This is using clearly expectation in the technical sense. The stock price may well go up, or go down, but we can't tell which or how much, so in the grand scheme of things, we're better off assuming it won't move at all.)
That's why you use a distribution to model the distribution of future possible prices given the information you have today.
The entire point of modeling the market as a martingale: you don't know what the future price is, but you do know a pretty good deal about where the price might go at various point in the future. Perhaps the single most important thing you know is that the future price is expected to be the same as the current price (ignoring the risk-free rate).
The Martingale property is very important to understand about stocks because if you are certain that the expected price of a stock is less than its current value you should sell, if you believe it will be more you should buy. An entire market thinking like this means that the current price should be equal to the expected future price.
Additionally these models don't "predict" future prices, but rather represent what the market believes about future prices given the current price and other information. This is essential to properly modeling risk and pricing assets.
To be clear, in a martingale the expectation is equal to the current value. It changes as the current value changes (i.e. as time passes).
>Especially when the first example is often the stock market, and everyone knows you can't predict movements of the stock market...
Not being able to predict movements is exactly what "the expectation is equal to the current value" looks like. If you had information that changed your expectation to be different from the current value, that would be predicting a movement in one direction or the other.
If my expectation for a stock is something other than the current value, I am claiming that I can predict the price. It doesn't matter whether my predictions are accurate or not, it's a statement about my beliefs.
A martingale model is consistent with believing that I can't predict the stock price. A non-martingale model implies believing that I can predict the stock price.