First, if money is a large object in the discussion, you may not be able to convince others unless it really is just a "little" more expensive than
RCCL. If the cost difference is significant, you may be out of luck. We've been on exactly two cruises. One with DCL with the grandparents and 4 kids, and one on Oceania, just dh and me. We loved Oceania, great service, great food, and I really enjoyed the people. No way, no how would I have brought my kids or my parents on that trip though. Not enough for either set to do. Our DCL cruise this summer was a complete success with the family, though. Everyone had a great time.
Consider how you, or the rest of your family, like to travel in hotels and resorts. If you consistently spend a bit more to get a superior experience (quality customer service, well maintained accommodations, clean restrooms, etc) then you may consider DCL "worth it." If some in the group (those that control the purse strings especially), prefer to feel that they've found a good bargain, are ok with adequate customer service, adequately maintained accommodations, then they may prefer another line.
We don't love Mickey Mouse, per se. My kids are 9-15, and are not into princesses and the like. We do appreciate good service though, and DCL consistently ranks high in that. We loved our DCL cruise and the kids are begging to do another one. That said, with most of the prices out there these days, we will not be doing much/any more cruising with them. Even we have a tipping point for cost vs "worth it."