Hmm, wouldn't the simplest explanation for the LM returning to the CSM before the TEI burn be for the simple reason of returning the two crew-members who went to the lunar surface to the CSM before initiation of the TEI burn?
Yeah, of course, but that doesn’t mean that was the only reason, and since that had to happen maybe TEI relied on the return of the astronauts and/or LM from the Lunar surface.
My point was just that if our only information is that the LM returned to the CSM and was jettisoned before TEI we can’t deduce that the LM was not needed for TEI.
Command module pilot Mike Collins, who could only keep orbiting in the mothership while his comrades waited out the countdown to lunar liftoff, worried enough for all three of them. “I have been flying for 17 years, by myself and with others,” he would write in his post-mission memoir. “But I have never sweated out a flight like I am sweating out the LEM [liftoff] now. My secret terror for the last six months has been leaving them on the moon and returning to Earth alone; now I am within minutes of finding out the truth of the matter. If they fail to rise from the surface, or if they crash back into it, I am not going to commit suicide; I am coming home, forthwith, but I will be a marked man for life, and I know it. Almost better not to have the option I enjoy.”
Two key sentence fragements: "leaving them on the moon and returning to Earth alone" and "If they fail to rise from the surface, or if they crash back into it, I am not going to commit suicide; I am coming home, forthwith". Both indicate that Collins fully believed he could execute TEI should the LM fail to return. The only way he could possibly "return[ing] to earth alone" should the LM "fail to rise from the surface" would be if the CSM could execute TEI without the LM ever returning from the lunar surface.
Which would indicate that the LM was not required in any way for TEI.
My point was just that if our only information is that the LM returned to the CSM and was jettisoned before TEI we can’t deduce that the LM was not needed for TEI.