Having done it both ways, I would not recommend the rendezvous diving if this is intended to be a family dive trip. All dive live-aboards would be happy to give you the whole boat if its available. There is a huge advantage to dive live-aboards vs charter yachts + dive op in that you can dive any time, any place you want. You'll have a dive crew, compressors, NITROX, rental and/or backup gear on board, at your beckon call without having to make arrangements or appointments for pickup. Its just waaay more convenient and much less hassle. Not to say a charter yacht couldn't have all this, but I personally like things more cushy than having to rendezvous.
A lot of people have recommended the BVIs, which I would too. Easy to get to, easy diving, outstanding cruising and sailing, and gorgeous topside scenery. Also great for families. The diving is not as good as many areas in the Pacific or Indonesia, but its a heck of a lot easier to get to, even from LA. I'm in OC and fly through San Juan, PR to Tortola. It's usually a red-eye and I'm there, usually tattered, the next day.
If comfort is important, Cuan Law is a great boat for what your asking. Not cheap, but worth every penny. They have 10 staterooms (hope you've got a big family) that can accommodate two people in each. By the way, I'm up for adoption if you need another daughter/personal photo-pro/tour guide. The food on Cuan Law is the best food I've had on any boat or at any resort world-wide. Number one, numero uno. The best.
Another family destinations you might like is Fiji. Only 10.5 hours non-stop from LAX on Air Pacific and you're in paradise. Boats like Nai'a are very popular so they are booked far in advance, but there are boats like Sere Ni Wai (now marketed by the Dancer Fleet) that is a small and very nice boat.
The Red Sea I have not been to yet, but its dropping further and further down my list as many friends come back with complaints of the northern areas being way too crowded, too polluted and no fish. I am considering the deep south. Boats there are amazing by most live-aboard standards and are not too expensive.
You might consider the diving skill level of your family before you make a decision. Not every destination offers easy diving and not all the easy destinations offer great diving. Perhaps your group is advanced enough to tackle some of the more challenging destinations, and then you can add Palau, Truk, Papua New Guinea, Raja Ampat, Komodo, Galapagos, or Socorro to the list. All these destinations can have ripping currents certain times of the year and might not be good if your family doesn't have a lot of experience.
Hope this helps.
Bonnie