What is the intuition behind linearity of expectations not requiring independence?

It's because summation and integration are linear operations: $$ \sum_j (a x_j + b y_j) = a \sum_j x_j + b \sum_j y_j$$ $$ \int (a f(x) + b g(x))\; dx = a \int f(x)\; dx + b \int g(x)\; dx$$ and expected value is defined by an integral (or a sum for the discrete case).

Here's an intuitive argument. Imagine that you repeat your random experiment $N$ times, each time observing a new value of a random variable $X$ and a new value of a random variable $Y$. Let's denote the observed values of $X$ and $Y$ as $X_1, \ldots, X_N$ and $Y_1, \ldots, Y_N$. If $N$ is large, then $$ \tag{1} E(X) \approx \frac{1}{N} \sum_{i=1}^N X_i \quad \text{and} \quad E(Y) \approx \frac{1}{N} \sum_{i=1}^N Y_i. $$ But $$ \tag{2} E(X+Y) \approx \frac{1}{N} \sum_{i=1}^N X_i + Y_i. $$ Comparing equations (1) and (2) shows that $E(X+Y) \approx E(X) + E(Y)$. And we can make the approximation as good as we like by taking $N$ to be sufficiently large. So we conclude that $E(X+Y) = E(X) + E(Y)$. Notice that in this argument we never assumed $X$ and $Y$ are independent.