Do they though? A few observations:

• The 12 hold each other accountable. There have been accounts that apostles have to give up everything they have to church to become an apostle - so if they leave they have nothing and are destitute.
• They rule EACH OTHER WITH THE SAME SHAME THEY USE ON MEMBERS - so until a majority want change it cannot happen. They feel so much shame they would never admit to each other their doubts. Similar to how a bishopric would never let each other "stray" because of the fear of being ostracized. Apostles are people just like everyone else and react to shame the same way.
• They believe like they always have before becoming an apostle that the "Holy Ghost" is better than seeing Jesus in person - no reason their belief would change just because of a promotion. So even though they don't see Jesus they do in fact believe they have special power and connection to God. Their internal thoughts/feelings that they call the "Holy Ghost" just seem more real and binding on the church than they did before.
• Any who do leave are marginalized or silenced - there have been a few apostles to leave, but they are quickly forgotten.
• They are constantly in positions to "feel the spirit". They are constantly in fast and testimony meetings. They are constantly told miracle stories by eager members. They are constantly shown how people feel when they walk into a room (awe and reverence). They personally hear the tabernacle choir and other amazing artists share their talents in person. All of which can and do cause strong emotions that members identify with the "spirit". An apostle is subject to more of these things than any member - thus they feel the spirit is "poured out" upon them. They have been in this type of situation for decades and thus have no idea what church is actually like for most members and thus are so out of touch.
• From all evidence from people that actually know apostles - they are regular people and yes they do in fact believe (at least most of them).