Do any civilian aircraft use AESA or PESA for weather radar?
Doppler and Electronically Scanned Arrays are different types of technology. Doppler is a method of using radar to determine relative velocity of a radar return. An ESA is a type of radar transceiver. Doppler is useful for detecting precipitation and would probably still be useful on an ESA radar.

Probably the main reason that weather radar has not widely gone to ESA is cost. Especially AESA (Active Electronically Scanned Array) is very expensive. In a military environment, the extra functionality makes up for the high cost. But current weather radar on aircraft works "well enough". The technology has been in use a long time and is well-understood. Advances in signal processing have brought more and more capabilities. There need to be large benefits to justify the expense of developing and certifying a new architecture.

One benefit of ESA radar can be the lack of moving parts. However, a single array would have a limited field of view, where current radars can typically see 180 degrees in front of the aircraft. They could add mechanical steering, but that would be even more expensive to steer something heavier and more delicate than the typical slotted antennas used today.

There's also the issue of military usage. Although many countries have developed ESA radars, there may be some government resistance to exporting that kind of technology.

Although current technology still struggles to detect hazards that lack precipitation such as clear air turbulence, the industry is looking to other technologies such as LIDAR to fill this gap.