Why don't docbots kill those who are awakened by the red pill?

Part of the purpose of the Matrix is to ensure all humans have an element of choice - the key principle in all of the Matrix films. The Matrix only functions because of choice. Humans have to be able to choose to reject the idealised, fictional world and somehow live in the real world. Therefore, as @SarpSTA states, some humans have to be allowed to escape for the world to function, otherwise the entire system would collapse.

However, having said all that, it's also clear the machine did try to kill him, and could have succeeded in doing so. It took a weakened, dazed human who had never truly exercised his muscles and flushed him into a giant pool of water where he was starting to drown. He had no means of survival whatsoever and only managed to live on due to the Nebuchadnezzar tracking the locator in the pill he swallowed and rescuing him. It's entirely possible that other humans in a similar situation may not be released in time, leaving them to drown.

Edit:

Just realised a similar question was asked over at the Sci Fi Stack Exchange: Why didn't the machine kill Neo after he was ejected?.
@Richard provided a great answer there, with some more information including an interview with Taki Kunitake, Head Illustrator for the film, where he states that the Wachowskis intended for the Doc Bot to have read him as a dead body and thus to simply dispose of him.

As far as I understand what is happening in that scene, the red pill fakes a death: From the script:

The mirror creeps up his neck as Neo begins to panic, tipping his head as though he were sinking into the mirror, trying to keep his mouth up.

NEO It's all over me --

Morpheus is right next to him, with the phone.

TRINITY I got a fibrillation!

CYPHER I knew it, I knew it...

MORPHEUS Shit! Apoc?

Streams of mercury run from Neo's nose.

APOC Targeting... almost there.

An ALARM, on Trinity's monitor ERUPTS.

TRINITY He's going into arrest!

CYPHER He's gonna pop!

APOC Lock! I got him!

MORPHEUS Now, Tank now!

His eyes tear with mirror, rolling up and closing as a high-pitched ELECTRIC SCREAM ERUPTS in the headphones --

It is a piercing SHRIEK like a computer calling to another computer

Neo's body arches in agony and we are pulled like we were pulled into the holes of the phone

Sucked into his SCREAM and swallowed by darkness.

I always thought that the pill simulated a heart attack (the fibrillation signal may be helping them to find the body), so the system just disposes him in order to be recycled. There is no need to kill a dead body.

It did try to kill him (or at least recycle him)

As a matter of fact, the Doc Bot did attempt to kill him almost immediately after it unplugged his ports, by flushing him down into a sewer where he would have drowned if there hadn't been a ship waiting for him. This seems to be standard procedure for any body that it reads as having been ejected from the Matrix (see below).

It didn't need to try to kill him (since it was already reading him as dead)

According to Taki Kunitake (Head Illustrator for the Matrix) in the The Art of the Matrix book, he states that the Wachowskis intended for the Doc Bot to have read him as a dead body anyway, presumably as a result of the pill that the Rebels gave him to "disrupt [his] input/output carrier signal"

Kunitake: One thing that was really nice in the film, something that wasn't in the storyboards, was the reflection Neo sees of himself on the face of the DocBot. I thought that was a really cool evolution from the boards, tying in to his whole path. When this robot comes down, very threatening. Larry and Andy wanted a bit of a shock, so it reaches out end grabs his throat. As an audience, you're thinking it's going to twist his head off. but it actually disengages. They'd talk about how the DocBot is reading Neo as a dead body, a dead battery, and disengages him so can flush him from the pod. Even in the script, they wanted it to be very graphic: very shocking. I was trying hard to get that sensibility.

Releasing humans is a necessity (since Zion is a sham)

Of course, we later learn that the entire Matrix/Zion project was set up to allow humans the illusion of choice. If your only options were to stay in the Matrix (and cause trouble) or to die, the system would break down much sooner. In that sense it makes far less sense to kill escapees than to actually let them escape.

Most probably in order to prevent any interruptions to the Prime Program since the One will be awakened with the red pill to start following the Path of the One.

Killing the One or those that'll motivate him with his ultimate goal of saving humanity will severely damage the Path of the One, probably causing it to fail, and drift Matrix into a catastrophe.