No, that has more to do with the material choices in the construction of the reactor. The moderator (the medium used to slow the neutrons to the desirable range for continued absorption but the fuel for additional fission) in different reactor designs isn’t always water. As I understand it, it’s the moderator which carries the temperature coefficient attribute. Water is negative, graphite (like in Chernobyl’s RBMK style reactor) is a positive temperature coefficient.

With a water-moderated reactor: temperature goes up -> total fission goes down -> power goes down (all else being kept equal). Which makes temperature come down. Which makes power go back up. This results in a sort of sine-wave oscillation of the reactor’s power level for a short time until other elements of operation come into play.

Graphite moderated reactor: temperature goes up -> total fission goes up -> power goes up. Which makes temperature go up. And the cycle repeats. This was a key oversight in design for what happened in Chernobyl, and why the choice of a water moderator helps to create a reactor design which is inherently stable.

Edit: Doppler broadening is more about why it’s preferable to operate with the fuel at a higher temperature from an efficiency standpoint in terms of using the available neutron flux to create sustained chain reactions.