DISCLAIMER: This is more hypothesis than definitive answer. I continue to research this, and will update the answer when and if I find new information. In the meantime, please don't let my "guess" dissuade anyone from answering definitively. I'll accept the first correct definitive answer, and my answer is not definitive at present!

Q1: Can anyone confirm that this is a hard and fast limitation?

Based on some material I've seen, AI Servo focusing is not available when using Live View on the Canon 1DX Mkii. This is a hard and fast limitation for the Canon 1DX Mkii: AI Servo uses 'phase detection' technology. The sensor utilized for phase detection in the 1DX Mkii requires that the reflex mirror be in the "down" position; i.e. positioned to direct light coming through the lens to the viewfinder instead of the sensor. Live View otoh, positions the reflex mirror in the "up" position. No light is transmitted to the phase detection sensor when the mirror is in the "up" position, and therefore the AI Servo AF subsystem can not function in Live View.

The 1DX Mkii does offer AF in Live View, but it is contrast detection AF, as opposed to phase detection AF. Note that phase detection AF has an advantage over contrast detection AF in that it is much faster.

Q2: Does anyone know the basis/reason why Canon might have chosen to disable this feature on their "flagship" camera model, yet enable it on a model that costs about one-fourth the price?

As @xenoid has pointed out, the "3rd mode" of focusing (labeled FlexiZoneAF() in the configuration menu, aka FlexiZoneAF multi) in the 7D Mkii is NOT labeled as "AI Servo". However, with selection of the Continuous Focus and FlexiZoneAF() options in the Live View menu, the 7D Mkii does provide what could be correctly categorized as an automated, servo-based autofocus system based on phase detection technology*. Not "AI Servo" perhaps, but certainly more functionality than offered in the 1DX Mkii!

Note* The 7D Mkii's Continuous FlexiZoneAF in Live View also utilizes phase detection AF technology! Canon has accomplished this on the 7D Mkii through utilizing one pixel of the "Dual pixel CMOS sensor" (imaging sensor) to perform the phase detection function. See this video for visualization. In other words, Canon utilizes the imaging sensor to perform both imaging and phase-detection focus measurement simultaneously in Live View. Note also that the 1DX Mkii also uses a "Dual pixel CMOS sensor"; the only apparent difference in the sensors being that the 7D Mkii sensor is smaller in size ("cropped sensor") than the 1DX Mkii's "full frame" sensor.

Perhaps compounding the confusion, the 7D Mkii was released approximately 18 months before the 1DX Mkii was released. One might easily imagine that this would have given Canon sufficient time to incorporate the feature into the 1DX Mkii.

Unless Canon has published an explanation, any answer to this question demands speculation as to motive. As far as I know, Canon has published no such explanation.