$\DeclareMathOperator{\Cov}{Cov}$
$\DeclareMathOperator{\Corr}{Corr}$
$\DeclareMathOperator{\Var}{Var}$
The problem with your line of reasoning is
"I think we can always assume $X$ to be independent from the other $X$s."
$X$ is not independent of $X$. The symbol $X$ is being used to refer to the same random variable here. Once you know the value of the first $X$ to appear in your formula, this also fixes the value of the second $X$ to appear. If you want them to refer to distinct (and potentially independent) random variables, you need to denote them with different letters (e.g. $X$ and $Y$) or using subscripts (e.g. $X_1$ and $X_2$); the latter is often (but not always) used to denote variables drawn from the same distribution.
If two variables $X$ and $Y$ are independent then $\Pr(X=a|Y=b)$ is the same as $\Pr(X=a)$: knowing the value of $Y$ does not give us any additional information about the value of $X$. But $\Pr(X=a|X=b)$ is $1$ if $a=b$ and $0$ otherwise: knowing the value of $X$ gives you complete information about the value of $X$. [You can replace the probabilities in this paragraph by cumulative distribution functions, or where appropriate, probability density functions, to essentially the same effect.]
Another way of seeing things is that if two variables are independent then they have zero correlation (though zero correlation does not imply independence!) but $X$ is perfectly correlated with itself, $\Corr(X,X)=1$ so $X$ can't be independent of itself. Note that since the covariance is given by $\Cov(X,Y)=\Corr(X,Y)\sqrt{\Var(X)\Var(Y)}$, then
$$\Cov(X,X)=1\sqrt{\Var(X)^2}=\Var(X)$$
The more general formula for the variance of a sum of two random variables is
$$\Var(X+Y) = \Var(X) + \Var(Y) + 2 \Cov(X,Y)$$
In particular, $\Cov(X,X) = \Var(X)$, so
$$\Var(X+X) = \Var(X) + \Var(X) + 2\Var(X) = 4\Var(X)$$
which is the same as you would have deduced from applying the rule
$$\Var(aX) = a^2 \Var(X) \implies \Var(2X) = 4\Var(X)$$
If you are interested in linearity, then you might be interested in the bilinearity of covariance. For random variables $W$, $X$, $Y$ and $Z$ (whether dependent or independent) and constants $a$, $b$, $c$ and $d$ we have
$$\Cov(aW + bX, Y) = a \Cov(W,Y) + b \Cov(X,Y)$$
$$\Cov(X, cY + dZ) = c \Cov(X,Y) + d \Cov(X,Z)$$
and overall,
$$\Cov(aW + bX, cY + dZ) = ac \Cov(W,Y) + ad \Cov(W,Z) + bc \Cov(X,Y) + bd \Cov(X,Z)$$
You can then use this to prove the (non-linear) results for variance that you wrote in your post:
$$\Var(aX) = \Cov(aX, aX) = a^2 \Cov(X,X) = a^2 \Var(X)$$
$$
\begin{align}
\Var(aX + bY) &= \Cov(aX + bY, aX + bY) \\
&= a^2 \Cov(X,X) + ab \Cov(X,Y) + ba \Cov (X,Y) + b^2 \Cov(Y,Y) \\
\Var(aX + bY) &= a^2 \Var(X) + b^2 \Var(Y) + 2ab \Cov(X,Y)
\end{align}
$$
The latter gives, as a special case when $a=b=1$,
$$\Var(X+Y) = \Var(X) + \Var(Y) + 2 \Cov(X,Y)$$
When $X$ and $Y$ are uncorrelated (which includes the case where they are independent), then this reduces to $\Var(X+Y) = \Var(X) + \Var(Y)$.
So if you want to manipulate variances in a "linear" way (which is often a nice way to work algebraically), then work with the covariances instead, and exploit their bilinearity.
