Decided to wipe it so I can give it out and... things went south really hard because built-in wipe in LineageOS borks the OS loader and no amount of flashing works. In the end I even borked the boot loader and things started to look really awry. I found a way to flash it, thanks to the guys[0] who found a sploit for MTK loader - and confirming what E4 was one of the most 'unkillable' phones of it's time.
But the hardest part wasn't that but:
actually finding a working copy of the official firmware
Google Apps really-really-really wants to update it all to the most current versions... and then it's just starts to fail every 5 seconds with an annoying modal notification.
Thankfully, F-Droid was already installed so I just disabled everything Google and made a... not-so-dumb phone. But, despite it working good enough - it's totally useless for anything other than calling and e-mailing with a 3rd party client (would K9 even install on it? I don't know).
Would it ever be possible to load a version of Android onto one of these phones with enough effort? I suspect the hardware would make it difficult.
Almost certainly not, and if you did it'd be practically unusable. The highest-end Sidekick devices had 128 MB of RAM; the first Android phones, like the HTC Dream, had 192 MB or more. It'd also have no network connectivity, as the Sidekick was a 2G / 3G device and didn't support 802.11 wireless.