Are drift current and diffusion current *momentary* in unbiased diode? Why drift current = diffusion current in unbiased diode?
When you form a pn junction, mobile carriers from each side of the junction flow to the other side due to diffusion. I believe you mistakenly called this drift in your question. Diffusion is the process by which things move from a region of high concentration to a region of low concentration. So in a pn junction, electrons from the n side, and holes from the p side, will tend to diffuse to the other side of the junction.

This diffusion of free carriers leads to a depletion of those carriers on each side of the junction. (Diffusion current) The depletion of electrons and holes leaves behind ionized donors and acceptors which gives you the charged depletion region. This charged region creates an electric field which causes charge carriers to move in the other direction. This component of current is the drift current: the current that flows due to the electric field near the junction. These two currents must balance out in order to satisfy KCL, and you have your diode in equilibrium.

There are cases where the currents are not balanced, such as if you apply a voltage to the diode, but in that case, the behavior is probably more intuitive. In your question you brought up thermal carrier generation. This plays a role in the diffusion current, since thermal generation is where the carriers come from to support the diffusion current. However, it does not grow out of control like you expect since it is limited by the drift current. If you were to increase the carrier generation rate, you would have either an increase in the potential between the p and n sides of the junction which increases the diffusion current to cancel out the increased drift current, or there would indeed be some net current if there is a circuit for it to flow through. This is precisely how a solar cell works. Photons create electron-hole pairs and increase the diffusion current.