The problem with fixed price work is you create a conflict of interests from the start. It's in your interest to do as little as possible for the price and it's in your customers interest to get as much as possible for the price.
That and all the issues of requirements gathering up front.
Isn't it also true that hourly or daily billing creates a conflict of interests from the start, because it's in the customer's interests for you to finish as fast as possible and in your interests to finish as slowly as possible?
Yes, but those are smaller than the converse, where the conversation is about whether something ambiguous/new is in or out of scope (strictly adversarial discussion), rather than whether such ambiguous items are worth it or not to the client (simply a resource allocation discussion).
I have a hard time believing that the potential conflict of interest situation that emerges from a fixed-price engagement, where the customer ultimately can accept or reject the work, would be worse than the elemental conflict of interest that emerges from hourly billing, where customers have little control or even insight into hours billed.
That and all the issues of requirements gathering up front.