Hello Avi
"Our students now work on a variety of platforms many with touch screens."
Moodle works really well on mobile devices (and has the free Moodle Mobile App)
"...side paging (opposed to linear reading)"
I'm not 100% sure what you mean here, but Moodle has side Blocks, and have you tried the Book and Lesson modules which can both have table of contents?
"H5P, ...can not do the job"
OK, that's a comment about H5P, not Moodle.
"...integration for all LTI capable systems"
Yes, Moodle can use LTI.
"Moodle core should be able use a drag and drop item system, to build a course schema."
Moodle does support drag and drop, as long as you are using a modern browser. I create courses all the time by simply dragging and dropping images, resources, files, etc. from the file manager window to the main Moodle course, it's easy
"Our great leap came in in Moodle 1.9."
I hope you are not still using Moodle 1.9? You don't say what version you are using? But if you are using Moodle 3.4, and you actually know how to use all the different parts properly, I don't understand what the problem is?