Sadly the High Sierra recovery server is still broken, but luckily this workaround works (just verified it today). (If your Mac is still under MDM you must ask the company to remove it, though).
First reset PRAM (just incase, cold boot and when hearing the chime start holding Option-Command-P-R until the Mac has chimed 4 times) and then follow the Fix#3 instructions to the letter:
https://mrmacintosh.com/how-to-fix-the-recovery-server-could-not-be-contacted-error-high-sierra-recovery-is-still-online-but-broken/
You can copy the magic nvram command from the Installer log and then slightly edit it as advised. I had some trouble finding some codes like "= from my keyboard but I could copy those from the Terminal (I guess I could have also switched the keyboard layout to match the physical keys before going to the installer). The final command should be as below.
Some other issues you might face:
AFAIR High Sierra might not recognize a volume after the 1st when installing. So if High Sierra is in the mix, it might have to be the first APFS volume.
As a test I formatted an external SSD device (not just volumes under it by choosing Disk Utility > View > Show All Devices) as APFS (GUID), but the initial boot seemed to stall. I reset PRAM again and Option-booted to the "macOS installer" and it continued from there. Maybe it would install more gracefully to MacOS Extended or maybe that was only a random hiccup?
After installing you might again reset PRAM to return to the default.
nvram IASUCatalogURL="http://swscan.apple.com/content/catalogs/others/index-10.13-10.12-10.11-10.10-10.9-mountainlion-lion-snowleopard-leopard.merged-1.sucatalog"