I think Doghouse traffic signals are better because you can turn on the green arrow and on the circular green. On a protected only left turn traffic signal you only turn on the green arrow unless, there is a flashing red arrow.
First, a clarification: The "doghouse" signal head ("5-section cluster" in a technical term) *IS* a left turn signal, just a specialized variant which allows for protected and permissive left turns (PPLT).
So the question which reflects the OP's intent is better phrased as "protected-only vs. protected-permitted". In this context, I think everybody would agree that a PPLT device (whether it is a 5-section display or an FYA) is often superior. The ability to yield to oncoming traffic during the permissive phase helps to reduce delays. PPLTs are not always the best option though; using a protected-only signal is warranted for certain conditions (i.e. multiple left turn lanes, sight distance issues, high opposing through speeds and/or volumes, etc.).
Doughouse traffic signals also can be use for straight through traffic .
Current MUTCD says a doghouse or 5-section display can only be used to control left and through traffic if the signal head is centered over the lane line. The circular green indication is not to be used in a signal face intended exclusively for the left turn lane.
Protected-only signals are appropriate on highways that have heavy traffic and/or multiple left turn lanes. The use of the flashing yellow arrow (FYA) allows, first, alternating between permissive-only and permissive-protective cycles at different times, depending on traffic conditions; and second, allowing permissive left turns when the through traffic in the same direction of travel is stopped for protected left turn coming the opposite way.
The PPLT benefits can also be achieved using a doghouse (or other 5-section display), as the Las Vegas area has used quite a bit over the last 5-10 years and Dallas has used extensively for many more years--hence "Dallas Phasing". It requires the doghouse be exclusive to the left lane, and have louvers shielding the circular indications from through traffic--back in the day, this also required quite a bit of reprogramming of the traffic signal controller. FYA operations make this type of PPLT mode switch much easier, and it's easier to program into the controller now.
And, you mention the flashing red arrow - how many locations use flashing red arrow versus yellow? I should note that, despite the proliferation of FYA in this area (CDOT District 2, southeast Colorado), there is at least one intersection in this area with a flashing red arrow at an intersection that has obstructed forward view. I should grab a photo of this unique installation, which features not two yellow arrows on a four-aspect vertical head like you see on typical FYA applications, but kind of an upside-down doghouse that offers two red arrows, one for flashing and one for consistently red.
FHWA approved a flashing red arrow in the 2009 MUTCD for just the situation you describe, where sight distance or other factors necessitate a full stop before a driver completes the permitted left turn. Due to this, it should be relatively rare. I believe one alternative signal head arrangement for a flashing red arrow signal allows for the steady red and flashing red arrows to be side by side but otherwise it looks like a typical protected signal--is this what you saw?