I spent a lot of time (almost 40hrs.) with this issue and I’m pretty pissed. But finally I solved it!!! It worked exceptionally only when I left my Omnia Box openned - it mean there was a ‘state: Connected’ in the Status field of the:
mmcli -m 0
And what helped? There was mistakenly connected middle antena connector of the mini-pci Quectel EP06-E card since Turris Omnia 4G manufacturing. The middle connector should remain unplugged, use only both on the sides one. Even of this issue, signal detection did not shows any issues, maybe because the right connector left properly connected, and mobile operators always accepted connection, but only in the enabled state (like in your case), but never falls to connected one (with one exception, when I left openned a metal cover).
and question regarding to ModemManager log:
logread | grep -i modem
Seems that there are some issues in the quality control during Omnia-4G manufacturing, hope that helps you too.