When a genetically-enhanced soldier escapes from a Military Black Site in New Mexico, his CIA handler struggles to find a way to apprehend him as the soldier leaves a trail of destruction throughout the United States.

The super-soldier is more interesting, however, you have to tell your own story,
The POV character does have to be the protagonist. The most obvious example I can think of is "The Great Gatsby".

The creator of "Breaking Bad" said something regarding framing a "bad guy" as the main character along the lines of: The audience does does not have to agree with their actions, but they do have to empathize with them.

Regarding the genre:
You can introduce the GMO through the CIA agent's POV and occasionally snap to his victim's pov doing fights.

To help with your logline:
POV: super-soldier
Event: escapes black site
Action: Free his brothers
(Story goal: to overthrow the US government)
(Stake: life or death)

This is my attempt, but keep in mind, I know a little about story structure and noting about loglines.

After his escape, a lab-born soldier struggles to free his brethren while his ex-handler follows the trail of bodies that will lead to their execution.

How should I reframe the logline with the GMO as the protagonist/anti-hero? I still think if it's a Military Horror, that looking through the CIA handler's eyes as the GMO is like this super boogeyman, wouldn't that make the GMO more scarier?

To overthrow the US Government and take siege of the White House. His adventure takes him to several Black Sites around the U.S. that contain other GMO soldiers so he goes into them to set them free so he can build an army.

Yes.

What is the GMO GI's objective goal?

Even if the GMO GI remains the problem instead of the protagonist, why is he on a violent rampage? All major characters should have an objective goal?

Hm, so with that in mind - does the logline need to be reworked to accommodate that?

I agree.

I think the GMO super-soldier gone rogue is the more interesting character. Suggest making him the pov character, framing the story and logline around him.

To clarify a bit also, what if Michael Myers was militarized?

It's grounded realism so the character should feel almost like what if the US Government produced a superhuman pretty much yet couldn't control him. Two main influences for this are Sicario (in terms of the realism), Brightburn and Akira. :)

The logline is clear, understandable, and sounds interesting.
If you are posting to test the concept, then my opinion is that you are ready to begin writing.
If you have written the story, the logline could be touched up.
The main protagonist (CIA Handler) is buried.
You could cut "from a United States Black Site in New Mexico."
Also, this has been done before, what is your spin on the story?