Here is derivation of drift velocity.

Assume there is a field $\vec{E}$ inside the conductor(wire). Using equations of motion we can say that for every charge inside the conductor, $$\vec{v_1}=\vec{u_1}+\frac{\vec{E}e}{m}t_1$$ $$\vec{v_2}=\vec{u_2}+\frac{\vec{E}e}{m}t_2$$ $$.$$ $$.$$ $$.$$ $$\vec{v_n}=\vec{u_n}+\frac{\vec{E}e}{m}t_n$$ where $t_1,t_2,...t_n$ are the relaxation times. Summing them and dividing by the number of charge particles($N$) we get, $$\sum_{i=1}^n\frac{\vec{v_i}}{N}=\sum_{i=1}^n\frac{\vec{u_i}}{N}+\frac{\vec{E}e}{m}\sum_{i=1}^n \frac {t_i}{N}$$

Since initial velocities are random $$\sum_{i=1}^n\vec{u_i}=0$$. Thus we get, $$\vec{v_d} = \frac{\vec{E}e}{m}\tau$$.

My question is why do we use relaxation time instead of using $\Delta t$ and then averaging the velocity as time interval for all electron's velocity in the wire?

As Wikipedia states "drift velocity is the average velocity attained by charged particles, such as electrons, in a material due to an electric field". So shouldn't we use change in very small time instead of using relaxation time?

Here's the link https://en.m.wikipedia.org/wiki/Drift_velocity#:~:text=In%20physics%2C%20a%20drift%20velocity,an%20average%20velocity%20of%20zero.