Until the point where your two proposed routes diverge in the middle of nowhere in Utah, you will for the most part only see mountains from a distance, rather than be driving through them. But the similarity ends there.
If you decide to follow the back highways through southern Utah and via Flagstaff, you will find yourself on very winding mountainous two-lane routes, at least intermittently, for hours, until you are well south of Flagstaff - which is itself in the mountains!
If you go via Las Vegas and Kingman, you will travel through far less of the mountainous terrain your passengers can't stand, though there will be some of it. In particular in the northwest corner of Arizona you will spend 15 miles or so passing through a mountain range, and despite being on I-15 the road narrows significantly in parts of it. You won't pass through any other significant mountainous areas on the remainder of the trip if you go this way.
If you really need the least disconcerting route for passengers who can't stand driving through mountains, I-15 through Las Vegas will be it.
You might also consider driving at night, so as to obscure the fact of the mountainous areas somewhat.