Not really. There are no "selfies" of voyagers, Dawn, New Horizons, Juno etc. T There ARE pictures/data from what these have produced. JWST will be no different.

Even Hubble didnt really have any of that that I am aware of. Most of the pictures of hubble itself are from servicing missions.

Sure you have some rover selfies on Mars, but thats not the sole purpose of the instruments, quite the contrary really.

How much electrical power do you want to devote to this feel-good PR selfie camera?

According to the mission director SmarterEveryDay interviewed on his YouTube channel recently, the James Webb Space Telescope is so sensitive it could detect the heat signature of a bumblebee at the distance of the moon.

I’m pretty sure if you asked the astronomers if you could put a 5 Watt webcam on the dark side of their sun shield for the social media clicks, they’d tell you to fuck right off to L3…

If it could see a bumblebee on the moon could it detect a while on Jupiters moons ? Can anyone do the calculations ?

It’s designed to see things very very far away, which are all severely redshifted. It’s mostly an infrared camera. In the human visual range of light frequencies, is camera sensors only go up to orange.

So no, it can’t detect a blue whale… ;-)

Well it can allegedly make photos of Jupiter that would be comparable of doing them during a close flyby, but that's likely not good enough for something of that sort. You'd have to see things through kilometers of ice.

I'm wondering if there is a correlation between the level of bowing to media hype by including cameras and the actual scientific value of the mission?

The recent filming of EDL of the Perseverance rover was spectacular, but a significant aim of that mission and indeed most of the recent mars rovers has been to inspire an interest in exploration and science in the public, a significant part of that is getting happy snaps of events as they happen. The pictures taken were also of significant interest to the engineers involved as they showed events that were extremely dynamic, video being one of the best methods to gain information about the events without impacting the operation.

Web is more aimed at direct science, the products of which will, as happened with Hubble be used to make pretty pictures, which in themselves have limited scientific value, but are of enormous value in engaging the public. This combined with the difficulty and risk of capturing meaningful images given the environment makes live video a non starter.