Lets be frank about this

David isn't really under the control of the company anymore, i highly doubt he would carry such mission.

Any space vessel under the command of Weyland-Yutani is suspect for experimentation purposes. In Prometheus, we saw Peter Weyland's obsession with creation along with his narcissistic personality.

The one thing that always throws me on this topic is the terraformers' shake-and-bake colony on LV-426 in Aliens. The families had been living there for years, and were never led to the derelict until Burke stepped in of course. So I'm assuming the majority of Weyland-Yutani vessels are created for legitimate missions. If the company's overall goal is researching the Xenos, they wouldn't want to make that known to society. This is why they only send vessels on the "crew expandable" missions every few decades or so.

Add-on: If David is experimenting on the Covenant crew, it has nothing to do with the wishes of Weyland-Yutani. He no longer considers them his boss.