Differential equations and LTI systems I've seen in many textbooks on Signals and Systems that an LTI (Linear Time-Invatiant) system can be described as a constant-coefficient linear differential equation, such as

$$\sum_{k=1}^N a_k \frac{d^k}{dt^k}y(t) = \sum_{l=1}^M b_l \frac{d^l}{dt^l}x(t)$$

Later on, it is stated that the response of any LTI system can be decomposed into two terms: a zero-input response, and a zero-state response: $y(t) = y_{zi}(t) + y_{zs}(t)$ (among other decompositions) which are independent.

Zero-input (ZI) response $y_{zi}(t)$ is the output of the system when the input is zero, that is, the output to the initial conditions of the system. This means that ZI-response is the same as long as the initial conditions are the same and no matter what the input is.

Zero-state (ZS) response $y_{zs}(t)$ is the output of the system when the input in nonzero and the initial condition are zero. Thus, ZS-response depends only on the input (via convolution integral etc.)

Now, the question: if the system is LTI, it means it satisfies the linearity and the time-invariance property. That is:

Linearity: for input-output pairs $x_1(t) \to y_1(t)$, $x_2(t) \to y_2(t)$, we must have $$ax_1(t) + bx_2(t) \to ay_1(t) + by_2(t)$$

However, this is not true for nonzero ZI-response $y_{zi}(t)$. If $y_{zi}(t) = e^{-t}u(t)$, for example, then this response is not scaled by $a$ or $b$ when an input is applied, because it does not depend on the input! Thus, any system with nonzero ZI-response cannot be linear!

Time-invariance: if the input-output pairs are $x(t) \to y(t)$, and $x(t-t_0) \to y(t-t_0)$, then the system is time-invariant.

However, since again the ZI-response is the same as long as the initial conditions do not change, there is no shifting of the ZI-response when the input is shifted! Thus, a system cannot be time-invariant!

This makes me assume that a system described by a constant-coefficient differential equation is LTI only when the zi-response is zero, that is, when the initial conditions are zero! The last one makes the system causal as well.

Where do I make the mistake??? It seems to me that linearity and time-invariance (and causality) is only valid when initial conditions are zero but this is rarely noted in textbooks...
Your "mistake" if it can be called that is to assume that your input signal was zero for $t\le0$.

When talking about the ZI-response and the ZS-response of a (typically causal) LTI system, "zero input" means "zero input from now on".

In that case, the past input's influence on the future signal is fully expressed with the ZI-response, while the future input's influence on the future signal is fully determinable by the ZS-response.