Anything beyond Cat I is usually just not worth the cost, training, paperwork and extra maintenance requirements, unless a private owner has a very specific repetitive requirement at an airport that frequently is below Cat I - and then only on a time critical basis.
Taking an AOC perspective, NetJets, the largest operator of business jets, doesn't bother with these qualifications.
Many private owners also can't be bothered to fly in poor weather - remember over 50 percent of people feel apprehensive at some level about flying. If going the next day, or later the same day solves the problem, then that's just fine with them. They are usually in control of their own agendas anyway, so rescheduling whatever they were there to do is usually simple and cost neutral - unlike a scheduled airline operation where delays drive cost explosions and close the loop on the business case for Cat II and III.
But there are always exceptions - I'm guessing that there may be a few ultra-longrange operators who regularly have places with naff weather as their destinations after a max range flight - they might find it useful.
Another example I have come across is that some owners want all the kit and all the qualifications - just because. But Cat II aircraft and crew qualifications are only exceptionally going to hold up as a business case.